You have to know what the watts required by the aerator is to determine the rest of the system. If you don't know that the rest is just guesswork and hit or miss. I know it all sounds confusing but it's really not that bad. i apologize up front if I get too wordy but that's just how I am.
Aerators generate about 1 liter of air per minute, or about 15.8 gallons per hour per watt.
So a 20 watt aerator would pump about 317 gallons of air per hour.
I'm just guessing, but I suppose a gallon of air would be equal to pumping a gallon of water per hour so if you use the guidance usually stated on this forum to pump a volume equal to your pond per hour, that 20 watt air pump should aerate a 300 gallon pond. I checked and found this size pump on Amazon for about $35 with an air stone and some hose.
Now, for the part I do know what I'm talking about, solar power and energy storage. I'll assume we are going to have a pump that runs on 20 watts and is on 24 hours a day. That is going to use 480 watt hours per day. If you only run half the day, then 240 watt hours.
A 12 volt, AGM, deep cycle, 90 amp hour battery will store 1080 total watt hours of energy and if you never discharge it below 50% capacity it will last about 7 years if charged properly. This is the most expensive part of the system at $169 with free shipping on Prime.
To charge this battery you need a 100 watt, 12 volt solar panel at $79 and a 20 amp waterproof charge controller for $20.99
( if you find a deal and get two panels or a larger one, I sized the controller to handle up to 240 watts total. Extra watts on the panels will help on cloudy days but just be sure they are nominal 12 volt rated. note: 12 volt panels produce ~18 volts)
Then, the final part of the picture, since that air pump I suggested is AC powered you'll need a 200 watt 12 volt DC to 110 volt AC inverter. I know it would be more efficient to keep everything DC, but the AC choices for pumps is huge and DC is tiny and much more expensive.
PS. I did find some DC pumps by searching for minnow tank aerators but I doubt it would be rated for continuous duty like an aquarium or pond pump would be. Would still be interesting to test. It would easily run on the battery above for more than 4 days for a 5.6 Liter per minute / 88 gallon per hour 12 volt DC pump drawing .46 amps, =~5.5 watts. This might work for a 100 gallon pond, or for sure two of them would handle a 150 gallon pond.