Can I use a 9-speed shifter with a 8-speed cassette?
Unfortunately you cannot do that. The COGSET and the SHIFTER must have the same "distance", or more correctly, the same lateral distance (on the pulley) for each lenght of cable pulled (on the shifter).

When you go from 7 to 8 speed, the distance between cogs is the same (because the freehub shell is longer), so you could use a 7 shifter with an 8 cogset and one gear would be unreachable;

But when you go from 8 to 9 speed on the shifter, the expected distance between cogs change (because the freehub shell for 8, 9 and 10 speed is the same). Then you need a 9 speed cogset too, so that EACH shift will correspond to the CORRECT DISTANCE of lateral movement in the derailer.

If you move from 9 to 10, the same problem happens. That's a perverse kind of programmed obsolescence, in my opinion, since even cranksets end up being incompatible due to the chain being narrower and narrower... (look for the Sheldon Brown "Nanodrive" system).

So the bottomline is: if you want to replace shifters, you'll have to replace the cogset, and probably the chain and cranks. I would not advise you to do so only for the sake of shifter change. Consider doing it only as a last resouce.

(I for one never changed to the 9+ speeds framework. I have 5 bikes in current use, all of them with the "7/8 speed framework", and I can easily find spare parts like cogs. My 7 speed shifter (a very old Alivio) still works fine, and it seems to me it is very rare for the shifter to have a poor working only because it got old).