I think it's a good appointment for them, but then again that's totally contingent on their expectations. If they're still overtaken by these illusions of grandeur that they're one of the big clubs then this will inevitably go south, most likely sooner rather than later. If they're willing to take it on the chin for a year or two and actually build something then I could see this having success.
Assuming he's married to the style of play he promoted at Southampton, he's going to need to completely rebuild their midfield. The problem with that is that they've already got a pretty bloated squad. Off the top of my head: Paulinho, Sandro, Bentaleb, Dembele, Eriksen, Holtby, Sigurdsson, Capoue, Livermore...just for the middle of the park. I'm not sure how many of those really fit his style of play, but I know for certain a couple of the more high profile names on that list don't. I think he's going to have to shift quite a few players out and we've learned the hard way how tedious that can be. They've already spent their big money and (seemingly) it didn't provide them with a single player their new manager will view as a foundational piece. That's a big problem. When we spent big and disastrously, we at least left Rodgers with a squad containing Luis Suarez. That minimized the work he had to do in reshaping our squad and he still had a hell of a lot of work to do. Pochettino has inherited an even tougher rebuild with fewer resources at his disposal. So for me it all comes down to their expectations. If they think this should be a quick turnaround because they've got a squad compiled with big money then they're just due for more of the same. I suspect that's where it's headed because I don't think Levy has a clue. If I'm wrong and they demonstrate a bit of self awareness then they'll be willing to play a longer game. I think they've got a good man in for that job. Not sure that's his remit, though.