Charging 12V/4A battery
A lead-acid battery labeled 12V, 4Ah and a lead-acid charger labeled 13.5V, 0.5A can work together, likely.  Lead-acid manufacturers prescribe that the battery should not be charged at a rate greater than C/3.  A 0.5A charge going into a 4A battery is C/8.  Expect that it would charge in 8 hours.

Nevertheless, take precautions.  Charge the battery outside at least first few times.  Lead-acid batteries (abuse-tolerant that they are) can vent hydrogen when abused.

Further reading: BU-403: Charging Lead Acid