Definitely listen to the advice not to do anything that hurts. You gotta be careful with shoulder pain, it's a very unstable joint and any problems can easily become chronic if not dealt with.
I'd be careful with the upright rows, they can actually cause further problems in people with existing shoulder pathologies. I'd recommend a simple movement such as laterals with the thumb pointed up instead (as long as they're pain free). Just until you can press without pain.
I can't tell much without actually seeing you, but it doesn't sound too bad. Probably an impingement of one the muscles of the rotator cuff, and thus inflammation.
Just a few balanced exercises for shoulder health should help.
Use a form of rowing that emphasises scapular retraction/depression such as face-pulls for something like 3x10.
Dip shrugs are also good for this. Set yourself up for a dip, but keep the arms locked and simply shrug the shoulders up and down. Emphasise pulling the shoulder blades down and back at the bottom.
A similar exercise but for a different purpose is the push-up plus. Just the same, get in the push up position and do reps pushing shoulder blades forwards and back with locked arms.
These exercises are important as they basically make space in the shoulder capsule by making sure the shoulder is sitting the right position.
Some form of external rotation would be good, preferably L-flies lying on your side.
It probably sounds complicated and lot of hassle but none of this is heavy training. It should be done with light weights for higher reps. Start about 20-30 reps and as you add weight bringing it down to sets of 10.
Once you've got the hang of it, it's easy to add in a few simple exercises a week and it will definitely help your shoulder health. They can be done regularly, at any time..the end of your training, 1st thing in the morning, before bed, whatever. And preferably a few times a week.
