Not quite. Almost all jets and the larger TPs are only certifed as multi-pilot aeroplanes.
To even start the type rating course for a MPA you need, as you say, a multi-engine IR, MCC and you need to have passed the ATPL theory. See JAR-FCL 1.250. This makes a lot of sense because the you have no chance in a course like this if you haven't done, e.g., any of the Class A performance work...
So in theory you could get a type rating for a MPA on yout PPL and get around 15 hours CPL course...
And you could not fly it as PIC, as this requires an ATPL.