You don’t need to memorize all 24x24 combinations (in fact there are much more, as e.g. India is UTC+0530, Nepal is UTC+0545, and various countries have daylight saving at different times, so London/Sydney has three different values throughout the year).
Just memorize the offsets from UTC+0000 (London in January).
Furthermore you don’t really need to memorize them as random data. India is obviously +0500 or +0600 or something like that, given its geographical position east. I don’t know what Japan is but I can guess +0900 just because it’s slightly east of China.
So if you want to use a memory palace technique, you could be as minimalist as having 3-4 rooms, each with a different setting/feature, and placing the country/city in the correct one.
So with 4 rooms, every place with +0000, ±0400, ±0800 can go in room 4, ±0100, ±0500 and ±0900 in room 1, etc.
Then when you think of e.g. Costa Rica, you remember it was in room 2, and from its location it’s clearly -0600 rather than -1000 or -0200.
You’ll need something to modify the half-hour offsets (and deal with Nepal…) but that is straightforward.
Finally, to get the difference between e.g. Costa Rica and Japan, you just remember Costa Rica is -0600 and Japan is +0900, and therefore the difference is 15 hours.
The trickest part is the daylight savings part since it changes through the year, but if you remember the wintertime offsets, then at least you can ±1 or ±2 hours in Summer to get it right.
Or if your method fails sometime, you can Google e.g. “time difference costa rica tokyo”.