You can do any of these variations for the trick to work. #1, however, is the only way to permanently unlock the bike.
Way #1) The main way to do this is to successfully complete all sixteen Crazy Box levels. This unlocks the secret bike, which can be used in arcade, original, and crazy box mode.
Way #2) While at the character selection screen, hold L + R. First release L, then release R. Again, hold L + R, then release both at the same time. Next, press A. If done correctly, you will now be riding a bicycle.
Way #3) Highlight a driver at the ''choose your character'' screen and press L + R, L + R, L + R, then hold L + R + Up and choose your driver. NOTE: Make sure you're holding straight up. Diagonal does not work.
Way #4) If you want to unlock the push bike in ''Another Day'' mode, first choose original mode. Then press R + L, R + L, R + L, R, R + A at the character selection area.