The original Lego Batman: The Videogame on PlayStation 2 does not include a dedicated "versus" mode. The game is designed primarily for single-player and cooperative story play, allowing two players to team up locally.
While there's no head-to-head arena, you can still create friendly competition:
Sequels Lego Batman 2: DC Super Heroes and Lego Batman 3: Beyond Gotham introduced proper competitive multiplayer modes, so if versus play is a priority, those titles are better options.