Can I cut my potatoes 2 hours before roasting or will they start to turn brown?

Chemically, potatoes do absorb water through diffusive properties. The rate at which this happens depends on the temperature and oxygen availability. You can soak your potatoes in cold water (around 35-40ºF) in the refrigerator with relatively few changes in texture, color, etc.

If you don't want to soak your potatoes, however, but want to keep them from enzymatic browning, I would suggest putting them in an air-tight container with a whole chopped onion. You can even nest the chopped onion in a smaller bowl if you do not wish them to be part of the recipe. The cut onion releases sulfides which bind free oxygen in the container, making it slightly hypoxic (lack of air). This has the same effect as soaking them in water as you are reducing their exposure to oxygen. The benefit is that you can keep them like this for longer, and the drawback is the onion taste that will be infused into the potatoes.

If you dont put them submerged in water, they will turn brown. Submerged in water, refrigerated, they will keep for days. Go for it.

Actually, presoakjng potatoes can help keep them from burning while roasting by affecting the starch levels (that's purposely vague cause I don't know the exact science behind it haha)!! I always soak my potatoes, or blanch them if I'm time-pressed, and they never turn brown.