"How do I know if it needs repotting and when do I do it?"
Grasp the trunk and wiggle it a bit. Is the tree well established in the pot or a little on the loose side? If it is well established it would be an indicator that a re-potting may
be due. Look under the pot at the drainage hole/s for evidence of roots. Roots growing from the hole/s indicate an established plant. While you are down there look for any wires that may be securing the tree.
Sometimes a tree will actually force itself up out of the pot as the roots fill the pot but you will not be able to easily judge this since you just received it. If you can remove the tree from the pot and have the root ball remain intact you can examine the roots. In fact if it does remain intact that is another indicator that the time for a re-pot is approaching. Take all of these possibilities into consideration.
In your climate there are two good time to re-pot a Chinese Elm. One is in Spring just prior to bud break on a dormant tree. Some prefer to keep this species as an evergreen and if you prefer this another option is to allow a brief rest period during the fall and early winter and re-pot before bringing it inside. I prefer the former scenario. Do you intend to keep it as an outdoor tree?
Also, I was given a drip feeder, which should be placed in the soil and lasts 15 days, is this needed, and how often?
I am not familiar with this device. Your tree will need to be fertilized regularly throughout the growing season but I am doubtful about anything that takes control away from me.