I've posted some tips for each mission's boss in the mission achievements here, and in the case of the later missions, tips for the whole level. Those are based on beating each mission on Very Hard without the Ulti Mondo costume or Dark Booster arm "infinite blood" upgrade, so check them out if you're struggling with a certain mission or boss.
Very Hard unlocks after you beat the game on Hard (or if you beat it on Normal, I believe -- not sure first hand because I did Hard mode first). As Quantum Binman says, you don't need to start a new game. All the story missions will then be available on the map. You can tackle them in any order. The achievement will unlock once you've completed all twelve on Very Hard, regardless of the order.
There is a definite increase in difficulty in the last few missions, but for most of the Very Hard play-through, it will probably seem a little easier because now you've got skills you've unlocked and your blood/health have presumably been leveled up quite a bit.
Unless I'm mistaken, you can play through on Very Hard using the "Ulti Mondo" costume (doubles your attack damage) and an upgraded "arm" that gives you unlimited Blood. They have to be purchased in the store, but they unlock after beating the game on Normal (or maybe just Hard - needs confirmed). If this is the case, it would make a "Very Hard" play through a whole lot easier, and some of the tips below (like conserving blood for finishing moves) unnecessary.
The downside is drastically reduced rewards for completing the mission, but that might not be an issue if you're struggling to complete some of the missions without those upgrades. Using those upgrades have an added benefit of helping achieve AAA ranks, but since that counts for any difficulty level, there's not much use in AAA-ing Very Hard mode unless you just want to try it.
Some additional tips:
Always remember to re-stock your 10-ticket supply of Mika tickets at the store between missions. A few times, I actually forgot to do it until my first revive in the next mission, where I noticed my supply was low.
When reviving, you can sometimes fail. But there's something that may help. You'll probably be tempted to start out mashing X as fast as humanly possible. Don't. It seems to me that the game wants you to mash faster as you approach 100%, so if you're really good at mashing X, you may be making things harder for yourself. Start mashing at about half speed until 50-60% percent, then get faster so that reaching 100% is a breeze.
Don't bother upgrading any sub-weapon (middle row on the upgrades menu) except for the orange gun, which you'll want to upgrade to the max. I rarely had much luck with any of the other sub-weapons, which are unlocked by finishing Gigolo missions in between levels. It's clear from playing through twice that these are optional, and the designers planned on people not necessarily even using them. It's possible that there are certain enemies that are susceptible to these sub-weapons, but I never found the need to experiment. You may want to upgrade other sub-weapons after you've upgraded everything else (there's an achievement involved for it), but my advice is to not rely on them during battle, and to try to avoid hitting RB by accident, which cycles between them. I did this quite a bit, but would cycle back to the main orange gun.
If you haven't already done it, upgrade "Healing" to the max. This lets you trade blood for health by pressing LB, and when it's fully upgraded, you can usually find a spare moment even in battle to distance yourself a little from the enemies (or the boss, as the case may be), and heal. This works best in areas where you're relying heavily on melee attacks that keep the blood meter going back up.
Auto Heal helps also, but it's slow and of no use in a battle. But if you're low on health and between battles, you can always just wait there for the healing to finish. From what I can tell, the last part of the meter will never fully fill in though.
When in battles against a lot of enemies, try to get your blood meter up enough to perform one-hit kills with RT + X. This works almost every time against the lesser enemies. If you're dodging and getting behind shielded enemies, it can work against them too. It will work against larger enemies at times, but it's best to simply thin out the other guys so you can focus on the big ones. The same tactic can be especially useful in the timed areas, since whittling away at their health the normal way will eat away at your remaining timer.
The big slow-moving brutes with clubs need to be taken out with RT + X, which means you need at least one blood rose filled. I often painted myself into a corner, using all the blood to heal or to shoot, meaning that even when I'd make progress against the brutes and get the RT + X prompt, there wasn't anything I could do. Filling up the meter by attacking them directly rarely worked. However, it seems that the game will place pots or boxes in the same area which will often have one or two units of blood you can find. If you're going to rely on shooting the enemy from a distance, though, don't break the pot/box or collect the blood until right when you're prompted with RT + X, since you'll waste it by shooting.
When there are floating machine enemies, try to take them out first with the main (orange) gun. They're easy, and can help you avoid taking unnecessary damage while you fight melee against the other enemies.
Where there are enemies shooting you from ledges or areas you can't reach, try getting some distance from the walking enemies and shoot the far-away guys for headshots. In some cases, if you're simply fighting the melee enemies and dodging a lot (B + left stick), you'll trigger a dodge burst against the snipers which let you do a one-hit kill with X against them.
For most battles, what worked best for me was to just get fairly good at dodging and slicing, without relying too much on the gun. Hit X a few times, move while pressing B, and continue. Most of the time you'll probably maneuver around them, which is great for keeping up the attack against shielded enemies. But sometimes you'll get a dodge burst (specially when surrounded by several of them) letting you mash X repeatedly for a second, rack up extra damage. I ended up pretty much needing to alternate X's and B over and over, to see me through many of the fights.
You can guard break with Y or guard smash by holding Y then releasing for a second (if upgraded). This works even at a short distance away, which can be helpful. However, don't both trying either one when you're surrounded by a lot of enemies. Keep up the slashing and dodging instead. However, when there are just a couple, of if you've somehow lured one away from the rest, guard smash can work well or mashing Y repeatedly can work against those shielded enemies. Once they're vulnerable, go to town with X.
The gold/purple ninja guys also need an RT + X finisher, so it's a good idea to keep a blood rose in reserve if you build it up and are in battle against one. Fortunately, it's a little easier to refill it with melee against them, than it is against the big brutes.