Once you find out the model Xbox, that will help you determine which monitor would work best. Since it is not a PC, freesync or Gsync is not important (AMD/Nvidia)
If it is just the original Xbox One, or Xbox One X, 60hz is fine for gaming. I game on my PC with a 60hz monitor and don't have any issues
Xbox One Series X or S(releasing November 10th has adaptive sync, I am not sure how this plays with monitors, but my guess it wont work due to the HDMI 2.1 requirement and the fact most monitors don't have that.
What I would be on the look out for is a monitor that has HDR, because all versions of the xbox support the enhanced color. If you are not going to go with an HDR Monitor you have a lot of options
- BenQ EW2780Q ($299) Is above budget, but wait for sale and you have a decent performing HDR Monitor. 350nits brightness will help with HDR Content @1440P. Not the best black uniformity, but overall the best bang for the buck.