I just use a plain disposable razor, the same as one I use on my legs. Every morning after brushing my teeth, I apply liquid hand soap to my upper lip, seems to work better than shaving cream or gel, run hot water over the razor, and just shave my upper lip. I don't find stubble to be an issue.
Once in awhile a little tweezing gets rid of a darker, coarser hair.
I used to use facial hair bleach, then when the hair came in coarser, tried a depilatory. But those actually took lots of time, and since not being used every day, didn't result in a consistently hair-free appearance.
One thing to note with a razor is to take care in not going diagonally at all, or you could get a nick.
Good luck in finding your solution!