Take your time after the G boss fight to get all the ammo along the walls before ascending; there's no need to rush at that point. Tyrant is a bit of a sponge and you'll want to make sure you have enough ammo.
To get the most possible time I'd recommend ignoring pickups along the way to the elevator. On Normal I was able to get by with a RGB mix, one first aid spray, and one frag grenade, so you don't need to be overburden your inventory with too many items. There's a typewriter next to the elevator alongside an item box so make sure you save in case you miss your mark.
His weak point is not his head, it is his chest.
You'll see an exposed heart right in the center. Smack that sucker with your bullets. As for dodging his attacks, it's pretty straight forward. Don't be stupid and keep your distance. After going to Red health pop your RGB herb and you can be riskier with your time spent shooting as it'll take 6+ hits to take you down as opposed to just 3.
When he roars and crouches he's pretty much begging to have his chest shot, to take your big opportunity there. Get enough shots in and his big attack will get canceled as he recoils from the shots. Be advised that he's charging up a move that will kill you with a lengthy, unskippable cutscene (the horror!) at orange health. I was unable to successfully dodge this move at orange health each time he got it off on me, but (I assume) running to the right might help some.
The rubble that falls will be an indicator of your progress in the fight. First two large pieces will fall, then another group will create a wall making the arena smaller, then you'll get the AT rocket. Blast the fedora tipper right in his smug face before the clock its 5:00:00 and you're gucci.
I was able to pop this without trying for it with 5 seconds to spare after doing quite a bit of fooling around pre-fight to try and nab all the pickupables which were admittedly unneeded. I only used the handgun and found that my ammo supply was perfectly adequate, but that was because I made a surplus of it using two big powders before my fight with G.