I have done this about 4 times now, the cost of changing the details is 50 euros and it is quite easy provided you have the right paperwork, NIE with current address on or an Empadronamento (paper from adjudamento to say you live here( they are more interested in the person buying so all their docs need to be originals the sellers docs can be copies, and all docs for the car, the website is www.dgt.es I hope you find this useful  :)
Whoevers name the car is in as of 1st Jan they will get the tax bill in May!