Hi Silver! I feel your pain! I'm in the middle of a rotator cuff flare up right now and am doing everything I can to make it go away, so I'm happy to share what I'm doing.
First, as with any inflammation, rest, ice and anti-inflammatories are standard therapies. Beyond that, the best way to help rotator cuff issues is a two part program of stretching and rotator cuff muscle strengthening.
Keep in mind that I'm not a professional in this field and am just passing on my incomplete knowledge gleaned from my shoulder surgeon and physical therapists. As background, I partially tore my rotator cuff in 2002 doing close grip bench press. In 2005, I completely tore it, plus ripped my supraspinatus and infraspinatus doing too heavy of an upright row. It was surgically repaired in 2006 and I spent about six months regaining use of my shoulder. Since then, the repaired side has continued to be weaker than the other but relatively pain free. About two months ago, I got cocky and bumped up my weights doing DB chest press and flyes and bam, I was right back into problems. I'm pretty sure I didn't rip anything this time and it's just inflammation; in fact, I can feel the tendon "pop" over the bone when I raise and lower my arm, which is a sign of inflammation.
At this point, I've stopped doing all other upper body exercises except for the PT, moved my mouse to my left hand, and do the PT exercises for an hour every day. I also take a prescription anti-inflammatory. It's gradually helping but I can see it taking another 6 - 8 weeks to resolve. All because of one stupid workout.

So results won't be fast, unfortunately.
Mel is absolutely right when she says that NO ONE should be doing upright rows or any exercise with internal rotation. It just places too much strain on the shoulder. There are plenty of safer alternative exercises.
So Rule 1 of rotator cuff rehab is no exercises with internal rotation (except for a few specifically designed for rehab). Rule 2 is to hold dumbbells in an upright position -- with thumbs up -- rather than horizontally. So if you're doing front or side raises, for example, the DBs are perpendicular to the floor, not parallel. This opens up the shoulder joint and reduces impignment.
A lot of us don't realize that standard shoulder exercises focus on the delts and ignore the tiny muscles of the rotator cuff and supporting muscles. The goal of rotator cuff rehab is to strengthen those little supporting muscles and to do that, we need to stick with light weights. Once we go over five pounds or so, the delts take over and that defeats the purpose of these exercises. So more weight is NOT better! In fact, even male bodybuilders are advised to stick to five pound weights for these exercises, despite them being able to bench 200+ pounds.
This is the routine that I've put together from all my past experiences with PT. I do a lot of the exercises as supersets because they're boring. I also mix in the stretches in between sets rather than doing all the stretches together.
I start with 60 Pendulum Swings, 30 in each direction - bend at the waist and let your arm hang down completely limply, make a circle about 12 inches in diameter. This exercise alone increases my range of motion.
Stretches - 5 each, 30 second hold
1. anterior bicep stretch - lace your hands together behind your back and raise as high as possible
2. cross body shoulder stretch - bring your elbow up and arm across your body, hold with your other arm
3. behind the back stretch - place your arm behind your back with your elbow bent at 90 degrees, grab your wrist with your opposite hand
4. doorway stretch - stand in an open doorway with your arms on the frame, bent at 90 degrees (goal post position). Lean forward to stretch.
Resistance Band Exercises - 3 sets of 15
1. superset of front and rear raises, internal and external rotation - fasten your band to a doorknob. Face away and do a set of front raises, turn toward the door and pull to the rear, turn sideways with your elbow at your side and pull from the middle of your body to the side, turn the other way and pull across your body to the middle.
2. lateral raises ("sword pulling") - stand sideways to the door and pull the band across your body from hip height to above your shoulder
DB Exercise - 3 sets of 15, use 3 - 5 pound DBs
1. Scaption, thumbs up and thumbs down - a raise with your arms halfway between a front and side raise. Do three sets with your thumbs pointing up and three with your thumbs pointing down, even though it feels awkward. Thumbs down is the only exercise that hits one particular little muscle (I can't remember which one).
2. External rotation - elbows at your waist, upper arms against your body, rotate out to the sides and back to the middle
3. Shoulder rolls, forward and back - arms holding DBs at your sides, all motion comes from shoulders like a shrug, except it's a complete circle in one direction
4. Reverse flyes
5. Front raises - remember to keep the DB vertical
6. Side raises - vertical DBs
7. I don't know the name for this one and it's the one I dislike the most, so it's probably the best one.

Stand with arms in the goalpost position (DBs by your ears) and rotate DBs forward until they're parallel with the
floor, then back to start.
I have to run right now so don't have time to google for illustrations, but if you don't understand what I'm talking about from my very limited descriptions, let me know and I'll try to find some pictures or videos. Best of luck to you and let us know how you're doing.