Not quite. The original versions of the game didn’t, but the Mega Man Legacy Collection has a selection of achievements, with several of them relevant to the first Mega Man. We’ve picked out ...
While this can make the idea of going through Card-en-Ciel dungeons a little boring, I feel the turn-based, deck-building gameplay actually improves upon the Mega Man Battle Network formula.
The recent Nintendo Direct also announced the release of several more ports for the Switch, including Persona 3, 4 and 5 and the entire Megaman Battle Network collection.