No Arduino required. All you need to do is float charge the battery, using a CC/CV PSU, that can supply >4A at between 13.5V and 13.8V, and is protected against loss of mains supply while connected to the battery.
An easy way to do that is to use a CC/CV (nonisolated) DC-DC buck converter module plus a second hand 19V 4.7A (90W) or higher laptop PSU to power it.
This module appears suitable:
https://www.amazon.com/Adjustable-Converter-Step-Down-0-8-28V-Inverters/dp/B07QB6577Nyou can find similar on Ebay, Alibaba, etc. Its important to choose one rated for at least double the output current you need as Chinese power ratings aren't trustworthy and you need it to be reliable.
Salvage the 12V rail Schottky rectifier from a dead desktop PC PSU (or buy an 8A or 10A, 20V or 30V Schottky diode) to use between the laptop PSU output and the module input to protect against loss of mains supply. Keep the diode on the heatsink it was fitted to. Set the module output voltage to be appropriate to float your battery (it depends on exact Lead Acid battery technology and ambient temperature - best to check with your battery manufacturer), then set the short-circuit current to be over 4A but under 10% of the nominal battery capacity in AH. The higher you set the current, the faster the battery will recharge after a power outage. The product of the current and the voltage shouldn't exceed 80% of the laptop PSU's wattage.
Finally you can hook it up to the battery, and use it.
You'll probably also want to add a low battery alarm circuit so you've got enough warning you need to find an other source of power if the power cut is longer than expected.
If you are using a flooded cell Lead Acid battery with removable cell caps, so the electrolyte can be topped up, it *MAY* be worth considering putting the charging circuit under control of an Arduino, so it can track the charge state of the battery and after a power cut, boost the voltage for more rapid recharging, cutting it back to the float voltage as soon as the battery is charged.