For what it's worth, If you only give him QQ's+ I have Flopzilla calculating ~34.5% equity.

If you include AKo & AKs, it jumps up to ~42%. (Becuase 9/21 combo's will be AK), giving you a lot of split pot equity.

KK+, & AKo/s == ~40% equity.

KK+ == 23% Equity.

So in my opinion yes, shoving is fine. Unless you have a read that he's never going to want to get it in with AK, I think it's a good play. Realistically, I expect a lot of villans at these stakes to show up even wider.

I understand how to calculate equity vs other hands but I was more concerned with how to properly calculate how often a shove needs to work preflop.

There is one mistake in the calculations.

The total pot size when AI will not be $52 but $46.33:

Villain's stack of $23.04 x 2 + $0.25 BB = $46.33

(The SB is not counted separately as you are in that position and have already paid it, so it's included in the call of his stack - or at an earlier point depending on what you decide to calculate. Can't find a way to explain that in better words it seems).

You can only calculate using effective stack sizes (the smallest stack involved, for two players), if you count your higher stack it results in a calculation that has you winning more than his stack when you do win, which is not the case. Against this player, you also have only $23.04 in play, so any calculation should use that as your stack size.

The final pot if you call him will be $46.33, and the amount you are calling is his stack minus your 3-bet size: $23.04 - $2.5 = $20.54. So you are risking $20.54 to win $46.33. Thus you need 20.54 / 46.33 = 44.33% equity for a breakeven call.

If you put his range as QQ+ you have 34.59% and should fold. (FWIW I don't think we can exclude AK for the typical opponent, as this sizing may well also include that hand).

This assumes that he never 4-bet/folds.

how to properly calculate how often a shove needs to work preflop.

You can use this formula, there`s other out there but this is the simplest one to me :

EV = F($Pot) + C(%W$W) – C(%L$L)
“F” stands for “times villain folds” and “C” stands for “times villain calls”.
$Pot = the size of the pot BEFORE you shove
$W = what you would win the times you get called and win
$L = what you would lose the times you get called and villain wins