Yes, bamboo is an ideal plant for growing in pots. Most bamboos are also evergreen. Many of them fan out quite nicely and would look good against that fence.
You wouldn't be able to hang bird-feeders from them. But you can easily support bird feeders in other ways. Birds are not too keen on landing on bamboo because it doesn't have any strong horizontal branches. But it's a good screening plant.
You just need to find a suitable variety.
Phyllostachys aurea seems to be very popular. It can grow quite large, but if planted in a pot, then that will constrain the growth.