"Why doesn't Mexico simply arrest Trump?"

There are a lot of economic ties between Mongolia and Russia. You don't get a lot of options when you're landlocked and sandwiched between two large powers. So, Putin has a fair amount of leverage there, more than in most places.

Some potatoes are simply too hot.

Their army size and military spending is a fraction of Russia’s, plus economic ties and nukes existing. Although now would be probably one of the best times to go against Russia militarily, since they’re stretched pretty thin at the moment, I doubt Mongolia would be the one to do it.

Because the potential cost to them doesn't out weigh the benefits (or potential costs) they get from adhering to the ICC regulations.

And then what?

Mongolia is fully landlocked and is a minor military power.

• They cannot get him out of the country to the ICC as neither Russia nor China would give them flight permissions. Russia for obvious reasons and China would surely not do it as well - as it would make travelling for Xi much more problematic, especially after what he plans with Taiwan.

• They cannot keep him there because the Russians, as military incapable as they are, would basically drive into UB and simply retrieve him. The Mongolian armed forces have a joint budget that is less than what a single CH-53K King Stallion costs. Their army is based on (hardly working) T54 and BMP1. Their airforce owns exactly two Mig29 Fighters (gift of Russia). And that does not even include the very real possibility that the Chinese do Vladimir a favour and get him - simply to make sure to keep the West occupied and shift the focus away from them and Taiwan again.

• If they keep him there and he dies...well....that would at least mean the end of the political ruling class of the country and could lead to the very real threat of large scale revenge by either Russia or Russia and China.

My personal prophecy for any scenario like that would be that a few thousand Mongolians die, around 300.000 end up in labour camps and Mongolia as a sovereign nation ceases to exist. Why? Mongolia exists mainly because two major powers "couldn't be bothered" to conquer them, it would make them look bad and a buffer to a not that welcome neighbour is always welcome. If Mongolia fucks it up for one or both of them it basically looses that "reason" to exist in the eyes of both the Russians and the Chinese. And they would be "bad enough" in the light of a lot of countries as well as their own population that a "swift special military operation" is justified.

because they do need him / fear him more than we do.

Arresting another country's head of state is about as clear a declaration of war as you can give.
So your question boils down to: "Why doesn't Mongolia go to war against Russia?"

I hope that all the signee states of the ICC treaty will call in their Mongolian Ambassador and make it clear in no uncertain terms that this might lead to other agreements and contracts with Mongolia might be at stake.