A new player arrived at my club in fastticker. At the same day, I got this message:

Richard Kleinjan skills have increased 5 times.

Sounds good, but I suppose it’s a bug?

Not a bug, that’s one of the effects of having “better youths” type of star players in your squad. They basically have two effects for new youth players: their base skills start higher (if you are not already a top ranked country) and they may improve one or more times immediately after joining, fast-tracking their evolution path.