I'm dusting off a 2002 mountain bike to use for commuting, and the existing old Deore LX 9-speed shifters are shot and need replacement. It looks like the current 9-speed Shimano options might be Alivio or Acera, but I wanted to make sure that they would be compatible with my old XTR rear derailleur. A newer derailleur might perform better than a 20-year-old XTR, but I would prefer to keep the classic one on there if possible. Will this new Acera with old XTR combo work ok for me?
Yes, the rear derailleurs are completely acceptable, the cable pull ratio did not change. You will not get any new features of SHADOW PLUS you get in new generations of XT.
A year ago I still saw XT M772 Shadow on offer, but it seems to be unavailable now. Deore M592 Shadow seems to be still available here and there. You might be interested in that instead of the lower Acera or Alivio lines.
A new derailleur might function better than a used 20-years old one, but you certainly won't get such tight tolerances, such good materials and such high-quality pivots in the low-end lines. Be prepared they may not remain in such a good condition, the old one remained to be, for such a long time. The pivots might become loose, for example.