Some points about your proposed setup:

• If you are using a SLA battery (missing in the block diagram), your UPS-based inverter may already have a low-voltage protection circuit, turning the UPS off. So, you may not need such additional circuit.
• If you intend to not use a LA battery, most probably the UPS circuit will not work, as most computer UPS are rated for 500W or more. The current drain requirements (for ideal 100% conversion efficiency), would need up to 40A at 12V. Assuming 90% Eff. at nominal rating, the standby by consumption may be 4A at 12V, most losses due to (that big) transformer. Adding that pump, another 20W would need ~2A at 12V. So, being optimistic, if it is able to start without battery (higher startup current), it would then need > 6A at 12V. But your PV panel is 50W (~4A), so clearly undersized and this setup may not run ok.
• Talking in terms of Efficiency, using a UPS inverter capable of 500~1000VA to drive a 20W motor source is an overkill and probably will not work properly in steady state: as your charging current may be about 4A but your running demand may be about 6A (said above). So the UPS will drain at 6A a charged battery (for instance 7Ah), the PV does not have capacity to fully recharge it at 4A max (sunny day), the undervoltage protection will trip off the inverter (e.g. 10V), the battery slowly recharges (e.g. 13V), and this short-On & long-Off cycle repeats.
• About the motor in your (aquarium/pond) pump: such pumps are basically a synchronous permanent-magnet motor, as they have a rotor with a permanent magnet and the field is then driven by the AC line. This motor load is highly inductive, while several UPS inverters were not designed (or intended) to drive such loads.
• About inverter technology: Depending if the UPS is a real/pure sine wave (PSW) inverter or a modified sine wave (MSW) inverter (some DIY experiments and videos seen here), the synchronous motor may work ok or not - I don’t have personal experience in that, but have seen videos where MSW inverters were used for compressor-based refrigerator (induction motor) and worked, while a microwave oven (HV transformer) did not work ok.
  My guess/bet is that the aquarium pump will run with a MSW inverter - but needs to be tested/checked both dry (quick test) and with water (under load). Surely comments from other fellows would be appreciated.

• A first point to check is to see if aquarium pumps with synchronous motors work “ok” with the MSW inverter - best guess for the “old UPS” circuit.
  For this test, you may have a similar pump and can power the inverter with any (car) 12V battery.

• If your UPS transformer is for high-wattage, it’s stand-by consumption will be higher. As you just need 20~50W (run~starting), maybe replacing the transformer by a smaller one (e.g. 12V to 240V 100VA) could make your setup run “continuously”, while you have enough sun.

• Another possibility is to try a smaller inverter, like those used in the cigarette socket and delivering up to 150~300W. Doing this it could be driven by your 50W PV panel.

• Undervoltage protection: I don’t know if they have under. voltage protection, but if not, you can Google for ready-made “over discharge protection” or “low voltage cutoff” modules, or check this answer with a simple cutoff circuit with simulation results.

In case you conclude that you really need a cutoff circuit for undervoltage protection - that I believe you don’t need it - there is a sugestion my final paragraph.

The reasons why you may not need the cutoff circuit but you may have other points to consider, then please follow reading.

Some points about your proposed setup:

• If you are using a SLA battery (missing in the block diagram), your UPS-based inverter probably already has a low-voltage protection circuit, turning the UPS off.
  So, you may not need such cutoff circuit for undervoltage protection.
• If you intend to not use a LA battery, most probably the UPS circuit will not work, as most computer UPS are rated for 500W or more. The current drain requirements (for ideal 100% conversion efficiency), would need up to 40A at 12V. Assuming 90% Eff. at nominal rating, the standby by consumption may be 4A at 12V, most losses due to (that big) transformer. Adding that pump, another 20W would need ~2A at 12V.
  So, being optimistic, if the UPS inverter is able to start without battery (higher startup current), it would then need > 6A at 12V. But your PV panel is 50W (~4A), so clearly undersized and this setup may not run ok.
• Talking in terms of Efficiency, using a UPS inverter capable of 500~1000VA to drive a 20W motor source is an overkill and probably will not work properly in steady state: as your charging current may be about 4A but your running demand may be about 6A (said above). So the UPS will drain at 6A a charged battery (for instance 7Ah), the PV does not have capacity to fully recharge it at 4A max (sunny day), the undervoltage protection will trip off the inverter (e.g. 10V), the battery slowly recharges (e.g. 13V), and this short-On & long-Off cycle repeats.
• About the motor in your (aquarium/pond) pump: such pumps are basically a synchronous permanent-magnet motor, as they have a rotor with a permanent magnet and the field is then driven by the AC line. This motor load is highly inductive, while several UPS inverters were not designed (or intended) to drive such loads.
• About inverter technology: Depending if the UPS is a real/pure sine wave (PSW) inverter or a modified sine wave (MSW) inverter (some DIY experiments and videos seen here), the synchronous motor may work ok or not - I don’t have personal experience in that, but have seen videos where MSW inverters were used for compressor-based refrigerator (induction motor) and worked, while a microwave oven (HV transformer) did not work ok.
  My guess/bet is that the aquarium pump will run with a MSW inverter - but needs to be tested/checked both dry (quick test) and with water (under load). Surely comments from other fellows would be appreciated.

• A first point to check is to see if aquarium pumps with synchronous motors work “ok” with the MSW inverter - best guess for the “old UPS” circuit.
  For this test, you may have a similar pump and can power the inverter with any (car) 12V battery.

• If your UPS transformer is for high-wattage, it’s stand-by consumption will be higher. As you just need 20~50W (run~starting), maybe replacing the transformer by a smaller one (e.g. 12V to 240V 100VA) could make your setup run “continuously”, while you have enough sun.

• Another possibility is to try a smaller inverter, like those used in the cigarette socket and delivering up to 150~300W. Doing this it could be driven by your 50W PV panel.

Stand alone Undervoltage protection - in case you still need one:
I don’t know if all (small/cheap) inverters have under voltage protection, but if not, you can Google for ready-made “over discharge protection” or “low voltage cutoff” modules, or check for circuits as in this answer with a simple cutoff circuit with simulation results.