Since you booked through a travel agent (lastminute.com) you need to go through them for any changes.

Name changes are often not allowed and it is very important to check before booking and once it is ticketed that it is correct.

Depending on the route, airlines and countries involved will determine whether you will be able to travel as booked.

I would try to get it changed through lastminute.com, but suspect you will not get very far with them. This is to be expected with most OTA's that sell cheap tickets. In future it would be best buying directly from the airline or from a real TA.

Otherwise, you can either buy a new ticket now or try your luck and hope you are allowed to travel. It may come down to the checkin/security staff on the day who may not allow you to travel since your first name does not match the first name on your ticket.