I am trying to book a flight that will be on an A320 NEO.
Does this have any 'new' systems or features, like the MAX MCAS system?
I'm asking this question also because one airline has just grounded it's NEOs...
Different airplanes, different systems; however, all modern airliners have computer assisted flight controls to some degree. These include stall protection of various types, of which Boeing's MCAS is one. Airbus uses different methods of stall protection, but there's still a computer involved.
Do you need to worry about the same issue? No. Part of Boeing's problem was that they didn't tell people about their system and pilots weren't trained. Had the pilots known about the system and how to disable it, the crashes probably would have been avoided.