Minecraft is suprisingly demanding, especially in later patches.
I used to play that on a Core 2 Duo equipped Macbook around 2010 or so. Even then it ran poorly with a low render distance. My much newer desktop struggles sometimes with minecraft, but it works a lot better than it ever did on my old laptop. I never noticed a difference in performance between 8x, 16x (default), and 32x texture packs on my systems, and optifine or other mods never made much of a difference either.
I agree with the earlier posts. Older games should run well. Old games from GOG should already be patched up so they can run well on newer operating systems, and old retail releases often have community patches or fixes to make them run, but not all games will run well without emulation (or at all). Some requires features that are not available in current CPUs and GPUs.
When I had an old or slow system, I'd mostly play SimCity 4 (limit the game to a single core, it crashes if you don't), The Sims 2 (this game requires fixes on newer hardware and operating systems), source games like vampire the masquerade: bloodlines, dark messiah, halflife 2, and other contemporary games.
Modern source ports of 90s games can run well too. Quake through quakespasm, yamagi quake 2, duke nukem 3d through eduke32, and doom, doom 2, hexen, and heretic through gzdoom (there are other engines like prboom+, but gzdoom supports hexen and heretic too).
Some newer games might run well too. Some of these might be a bit sadistic/hard tho, and they're not all that new. Dustforce, Devil Daggers, Torchlight, Banished, Cultist Simulator, Darkwood. Banished is definitely a great game if you like something more comfy and relaxed.
Some of the more retro looking (new) games have surprisingly high system requirements, so I don't know how well they would run on your system. If I knew exactly which Celeron you had, it would be easier to tell.
Never pay full price for games (now is the perfect time, GOG has a sale right now, and steam will have one starting in a couple of days or so). :)