Most of these are self-explanatory and cumulative. You don't need to somehow spend 30 minutes straight in roll back mode, of course. The challenges are as follow:
1) Promoted to captain (previous rank)
2) Story 1CC (Complete story without using a continue)
3) Use the roll back capsule for 30 minutes
4) Clean game time is over 15 hours
5) Use the speed up capsule for 20 minutes
6) Collect every score token through 3 stages
7) Complete a stage utilizing the extend power-up
8) Immediately recollect 9 primary weapon power-ups
9) Destroy Siriad
2) "Story 1CC" can actually be completed simply by loading up a late part of the game and doing it from there. You can just load up the last boss and finish it off, as confirmed by Petit Ver.
3) & 5) If you're anything like me and my friends, you'll have every other challenge for both this achievement and the next one completed before you even have 10% of the capsule requirements. The best way to boost it is to go into Boss Training, pick any boss, Normal difficulty and with everything set to max. The reason for picking Normal is that you get more usage out of your capsule bar on that difficulty, meaning more time on the clock for your achievement. Just let the level start, hold RT until it runs out, pause and restart stage. This is going to take hours and you'll be wishing they put these challenges on one of the last achievements instead.
6) "Collect every score token through 3 stages" only refers to tokens on screen, and doesn't imply that you need to kill every enemy. This would most easily be done in Story Mode on normal, as there's less other things to worry about. By score tokens, they mean the round icons with numbers on them. I would assume that refills for your subweapon, time and capsule don't count. Be wary of enemies very close to the edge of the screen - don't even shoot at them if you're not certain you can pick up anything they leave behind.
7) "Complete a stage utilizing the Extend power-up" is an ambigious one that I can't fully define yet. The Extend power-up is a randomly dropped token like any other, and when you have one the word Extend lights up under your time at the top of the HUD. Wether they mean for you to use it before finishing a stage, or to complete a stage with the word lit and not using it is unclear. I would assume it's the latter since that's harder, but that's not what utilize means. Clarification in the comments would be appreciated.
8) You can quickly and easily complete "immediately recollect 9 primary weapon power-ups" by starting an easy boss in Boss Training, setting your power to 10 and ram into something immediately after starting. Then just pick up the 9 powerups and kill the boss.
9) Siriad is the name of the final boss. So, basically, this is a challenge for completing arcade, regardless of the amount of continues you use.