I have to concur with @dverhey on this one. The Mac Mini has a worse GPU (GeForce 320M) than even my non-gaming Toshiba T135 Laptop… which stands to reason since the 320M is made for non-gaming laptops. This is compounded by the fact that you cannot upgrade the GPU, so unless Minecraft is the absolute most intensive game you will ever play, the Mac Mini is not a good choice, especially not for the price. You would be better served by a PC or the latest iteration of the iMac. (The earlier generations of iMac suffer from the same flaw; a laptop GPU. Only the latest ones use a chip suitable for more than the most casual of gaming.)
If you insist on a Mac Mini, you don’t even really need a monitor. I am using a 32” 1080p flatscreen TV that I got on sale for not much more than you are looking to spend on a monitor alone. Given the ubiquity of HDMI ports on newer TVs these days, that is a viable option.
As far as quality goes, pretty much any screen that doesn’t suck is pretty decent and usually indistinguishable from any other non-sucky screen except by the most finicky people like @johnpowell so I wouldn’t worry about that so much, especially not if it’s paired to a sub-par GPU.