1.) I speak no Spanish. None. I can, of course, jot down a few phrases, and I’ll be sure to learn “please,” “excuse me,” “thank you,” “yes”, and “no” in Spanish, but that’s about it. Can I realistically expect to get by in San Juan?
San Juan is the capital. It’s the biggest municipality and main part of the metropolitan area (área metro). Which part of San Juan are you talking about? In Old San Juan, I’d say yes, for sure, you can get by in English. If you go to any tourist attraction or museum, yes, there will be English-speakers. If you want to venture to less touristy areas… Well, their English is probably better than your Spanish, but how well each other will be understood may vary.
2.) Can I get by without renting a car?
NO. NO. Not unless you are using a tour company or scheduled tours and then they can arrange travel for you. If you were better in Spanish, I’d say try giving public transportation a shot, but nope. Really… Just… no… Taxis are very expensive, you’re better off renting a car.
The only place I can make an exception is if you stay all your time in Old San Juan. Then you certainly wouldn’t need a car, as everything is within walking distance. Well, you may have to walk a lot, but it’s not a killer walk from La Fortaleza (Governor’s mansion) to El Capitolio (Capitol).
3.) Can anyone recommend a good cheap hotel?
In San Juan touristy area, nope. I know there are some places, but I’m not exactly sure where or how to get the contact information. I’m from San Juan, my family lives there, I don’t need a hotel. 
I’d say, and this is a born and raised city girl: GET OUT OF SAN JUAN!!! My island is beautiful, it is precious, and it is so much prettier outside the city!!! The best beaches IMHO are outside, food is better (and cheaper) outside.
There have been other recent threads about Puerto Rico… I’ll try to dig them up.