From what I could find Charlottesville is zone 7, so you have lots of choices, but most (all?) of them are running bamboos. Containing their spread with a barrier of some type is a good idea, but sinking containers probably will not work -- the rhizomes will escape through any drainage holes that the pot has. You should be able to leave the pots above ground though.

You'll only want to put one plant in each pot, as each plant might have several culms (shoots, canes) already but will definitely produce more each year.

The problem with growing in pots is that the plants will get "rootbound" after a couple of years and will need to be reduced by division. You could instead put the plant(s) in the ground and use rhizome barrier, or rhizome prune to keep the plant's spread controlled.

Burying the pots will NEVER ever keep them confined. They will eventually burst though the plastic or rhizomes will find another way out.--Bamboo roots would probably get though a concrete planter eventually! As suggested above, if you go with bamboo, you are either going to have to rhizome prune once or twice a year, or have a rhizome barrier put in.