That's going to be a Moentrol valve (because you pull to make the water run and push to shut off, and can control the water volume by how far out you pull).
The valve bodies on these things, as Standardairconditioner said, can last basically as long as the house will stand, but the cartridges won't. Since the thing is dripping, I would change the cartridge, which, as SA also said, you can do anytime from the front.
As far as I know, all the Moentrols use the same cartridge (different from the cheapo Positemps). When you buy a new cartridge, it comes with a little cap that makes it easy to remove the old cartridge. It's a pretty-easy, straightforward job.
And the cartridge is inexpensive, and it's the thing to replace if you have a leak. Well, actually, there are two cartridge options, the same design, but made of different material. One is the brass one, Model 1200, which has a list price of $48 but can be found for about $28. The Model 1225, which has some plastic components, lists for $29 but can be found for about $17.
I would spend the extra $10 to get the brass one, but the 1225 is the biggest seller, of course.
Were I you, I would buy the cartridge and replace it. I would clean up the trim with a Mr. Clean Magic Eraser, which is amazing on tile and shower trim, so it looks nice and shiny. Your tenant will think you just put in a brand new valve, and you will save water from fixing the leak.
(On tile or tub with lots of soap scum on it, I spray first with Scrubbing Bubbles, then wipe with the Magic Eraser, and the stuff comes off like it was never there. After years of using other means, this is far and away the easiest and most-effective combo to make the stuff come completely-clean with no elbow grease at all. Amazing.)
Here's a pretty-good video about how to change the cartridge. It really is easy and straightforward:
Note to nerds like me: I like how, in the video, when he pulls out the old cartridge, there is a piece of water heater dip tube riding on top.