Can I take admission to Presidency University, Kolkata (through PUBDET) if I get admitted under Calcutta University in the first year?

You certainly can. If the registration process is completed by you already, make sure you take your TC and mark sheet from there. Your mark sheet should have an "admission canceled" stamp to it otherwise it would not be approved. You may also be required to write a letter sent to the authorities. It will cost you a sum of ?1000-1500 which would have to be deposited as a TC fee.

In case you have missed your registration, make sure you get in touch with the staff of the office. They will help you get your admission canceled.

Although this is a rare situation, in case you are lucky, they may give back your admission fee too.