Hold L1 + R1 at the character selection screen. Release L1, then release R1.
Hold L1 + R1 again, then release them simultaneously. Then, press X. 
A ringing sound will confirm correct code entry,
and your character will mount the bicycle.

Alternately, highlight a driver at the character selection screen, then
press L1 + R1, L1 + R1, L1 + R1,
then hold L1 + R1 + Up and select your player.

Alternately, successfully complete all sixteen Crazy Box levels to unlock
the Secret Push Bike for each character. It can be used in Crazy Box, arcade, 
or original modes.

To unlock the Push Bike in Another Day mode, select original mode.
Then press R1 + L1, R1 + L1, R1 + L1, R1, R1 + X at the character selection
screen.