I need to add a UPS to my power supply circuit
60s * 5V * 0.5A = 150 Joule. You need more than 2F capacitor @ 12V =&gt; impractically big. No capacitors, then.
On the other hand, a couple of these:
https://www.google.com/search?q=3.6v+nimh+rechargeable+battery+20mAh

packs ready to solder on a PCB
two of them 3.6V each so 7.2v in series, enough to get 5V out of your regulator, it will be good for more than a minute (and much more with better regulator)
absolutely tolerant to overcharge so the charging and switchover circuit is 2 diodes and a resistor. Constant charging current of 0.2-1ma is acceptable.
NiMHs work from ~ -20 deg Celsius up,
NiMHs live longer than lead-acid battery

Be aware that low-quality ones leak a bit after a prolonged use/