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.
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.