Pushups. Unless you're completely feeble or injured, do them, even if it's on your knees. Guaranteed to clear up the majority of back pain due to weak muscles.
Also Yoga, I picked this up recently for a low impact workout. That shit will change your life, WAY harder than it looks.
Yes doing push ups with proper form will strengthen your core, but a lot of upper back pain is caused from tight pecs and lower delts which cause a hunch. Doing push ups would continue to tighten the pecs and contribute to back pain. They'd be counter productive for upper back pain, really.
Most people would probably get more back pain relief by laying face down on the floor for 10 minutes rather than doing 10 minutes of push ups.
Most people suffer from lower back pain, example all of us sitting here days on end at the computer. Obviously for upper back pain, working your upper back is not a great idea. My point here is that most back pain is related to our sedentary lifestyle, and simple pushups alleviate a lot of it.
Pushups work much more than just your chest, particularly your core muscles, critical to everything including your back. Try just a static plank for ten minutes, and tell me what you feel.
While I agree your core is critical to your back, a pushup or plank also strengthens your hip flexor muscles. A couple of those (e.g. psoas) attach to your lumbar spine, so strengthening them makes them shorten and pull harder on the lumbar spine, increasing your pain. For low back pain, it is the wrong exercise.
For my low back pain, I stretch the hip flexors to relieve their pressure on the lumbar spine. I strengthen my abs with side planks to stabilize my core without engaging the hip flexors. Finally, I strengthen my glutes.
Every situation is different, and not all back pain should be treated with a pushup, so you have to understand body mechanics so you're working on stretching or strengthening the right parts for you. Going to a Physical Therapist is very helpful, then progressing to some appropriate strength training.
Also Yoga, I picked this up recently for a low impact workout. That shit will change your life, WAY harder than it looks.