So you want to go from not enough Japanese to order food to passing the EJU in 1.5 years? Good luck man. You'll need it; it's supposedly really, really hard. Just stick with it and you should be OK.
1. The "easiest" would be the taxi, but it's going to cost like $400 USD to get from Narita to Shinjuku. There may be a bus as well, and these are a good option if you have a lot of luggage; taking all of your stuff through the train stations (especially huge, crowded ones like Shinjuku) can be a hassle.
2. There are a heap of places to eat in Tokyo and you can be sure that most places in Tokyo have at least some experience dealing with foreigners. Most places that have English menus will advertise this fact somewhere on the storefront. Alternatively,
English OK lists a few places.
Two other good options for foreigners are quick service-style places (which may or may not have seats) and convenience stores. The quick-service places are very easy to identify because they often have ticket vending machines. You just insert some money into the machine, push the button for your selection, get the ticket, give it to the person at the counter, and wait a couple minutes for your food. But keep in mind these vending machines rarely have pictures of the items, so you'll need to be able to read the text on the buttons.
Convenience stores have a lot of different kinds of boxed lunches/dinners and they're usually fresh especially around the regular times. Just pick whatever you want and go up to the register. Depending on what you got they may ask you if you want to have it heated ("暖めますか？ / attatamemasuka?"), so unless you want to eat it cold/eat it later, just say yes.
3. I don't have glasses so I can't really speak from experience, but there are a lot of different eyeglasses shops in Tokyo and I imagine they could handle most simple repairs. I don't think you'd need to goto an actual doctor unless it had something to do with your prescription. Take a copy of your prescription with you.
4. Walk is fine as Tokyo isn't too big, but stay aware... especially around Kabuki-cho. I will venture to say that even Japan's red light district is safer than most places in the USA, but that doesn't mean you should completely let your guard down. If you walk through Kabuki-cho during the evening hours you'll encounter a lot of people trying to sell you their "services," but if you just keep walking and ignore them they give up quickly.