There is one massive issue here that I've seen (and done) 100s of times - you've tried but then failed to talk to your SO about the cost of stuff. Then, even if you are constantly thinking about it, he isn't. So when you come out of left field (in his mind) pissed about money, he's taken back by this sudden attack. This is a communication issue plan and simple. Luckily, you've been able to spot it and can help fix the problem.
My thoughts - you're on the hook for the $1300/month for rent. While it might be more than you are wanting to pay, and seems unfair to me (he makes more, lives closer to work, picked the place, and pays the same?) you've agreed to it so let that water flow under the bridge. If it was me - start refusing to go out to eat with your SO. If he wants to go out for dinner, say no thanks and start cooking. I'd be surprised if he goes out more than once without you. When he asks why, just says you don't feel like paying for a meal when you are paying to rent a full kitchen. Invite your friends/other interns over to your apartment, sounds like you have a nice apartment, probably better than most of the other interns, so host some Friday night drinks at your place - still get the going out/friendship but for a faction of the cost.
Take this summer as the chance to talk about long term goals. you are living with this guy, so I'm taking it as it is serious with looks towards the future. Make sure you both are looking in the same direction (love that quote). Don't be afraid of upsetting him, because it is better to do it now then to do it in 2 years, or 5 years, or never getting to live the way you want. Take this summer as a way to 'stress test' the relationship and see if you can deal with the true person your SO is (and if he can deal with you). Plus, maybe he'll like the idea of being able to leave the high stress NYC job market after 5-10 years and do his work anywhere since his paycheck amount won't matter (I find that younger people don't understand the whole hate your job, but do understand that there are stressful and low stress jobs - with a paycheck being one of the big differences).
Understand compromise. Assuming long-term you two stay together, you are looking at 180k/yr - probably more b/c interns usually get paid less than normal employees. So you are going to be able to have a ton of savings while still keeping a very "MMM-spendy" lifestyle. The trick is to get something for each of you. If either side feels like they won, you both failed. If he wants a big 2 week vacation in Europe, you want an equiv amount of index funds - and both paid for before the plane lifts off; plus both sides benefit, you get the same vacation as him and he gets the same FI as you. He wants a new car, you want to max all of both of your retirement accounts. He wants to start becoming a wine coinsurer, you want to eat home 6 nights a week. He wants cable, you want a weekly trip to the library that he comes with you on.
