Not cooling could be :
1) Blower (indoor unit) wind not strong/not blowing -->indoor capacitor RM 3 (can diy)
2) Check compressor (outdoor unit), if ice building up at vapor line --->. need to recharge gas (need handyman to repair leaking & charging)
3) Compressor or fan not starting up --> investigate & start checking fan & compressor capacitor (can diy)
4) A/C cool for first few minutes --> Fan problem and/or compressor overheat heating (need handyman)

** if your compressor not blowing hot and strong wind, you have problem with A/C**
To check leaking, use soap mixed with water will do. No need spend extra $.

If leaking via the connector, try tighten the nut. If still leaking, either you call an AC handyman.
No point DIYing if you only have 1 or 2 units, because you need to buy many tools :
-  flaring tools & cutter RM 50-90
- manifolds gauge RM 130
- R410A gas 13kg RM 400 (can use >10 times though)
- vacuum pump RM 250-350
- Clamp meter to check amp draw RM 50 - 200

It's normal the compressor (outdoor unit) will start after 10sec - 4min), and it depends on brand and model because parts are different such as PCB, coil sensor, temperature sensor, etc.