If you're immediately sent into the first mission, quit out to the dashboard if you want to restart on Hardcore. Otherwise, continue through the game, blowing everything up as you go. I honestly didn't find Hardcore too difficult, as long as you're always staying in motion, you watch for grenades, and you manage enemies properly.
In order to survive in Hardcore, you'll need to utilize the roll ability (), and you'll probably want to focus on upgrading your shotgun first, as it does massive damage, has great range, and a really wide spread when it's maxed out. You'll also want to learn to utilize the special ammo on certain enemies (ie use armor piercing ammo against minigunners, homing ammo against flamethrower guys/guys up in rafters, and shock/incendiary ammo against pretty much everyone else). The greatest threat to you in this game, aside from mortars, are the melee guys that run up and cut your face off with their machetes, which is where rolling comes into play. Another tough adversary is the shotgunner, as they generally come in groups, and a single blast at close range from even one of them will take you down at least 2/5 of your health.
For the final boss, you will want to use your best weapon (again, shotgun is money here. It will drop about 3% of his health per hit at close range) and focus on him as much as possible. You can utilize homing and armor piercing bullets to quickly cut down the other guys that attack you in the mean time. One thing to note is that you will want to try and fight him where he first appears, because if he drops below 75% health in an area other than right above the elevator, the second phase of the fight will start. Try and get him to around 66%, then move to the middle of the submarine, and backpedal back to the elevator you both came out of. Once the elevator with the three guys comes up, take them out and grab the grenade launcher to deal about 63% damage to the boss. The remaining couple percent go far slower than the rest so just stick with it and you'll come out victorious.
Glitch note: Save before using the elevator to leave the submarine as the boss can glitch if you happen to die when he does, making him completely immune to damage when you reload the checkpoint. Having a handy save before the fight will save you a headache.