Not-quite-opposite of "inherit", someone from whom you inherit without their mentioning you in a will [closed]

The word you mentioned

bequeath

is the correct verb for giving things in a will (after you die). The person who does the giving is called a

benefactor

(though that works for people who are still alive). The thing that is given is a

bequest

Putting it all together:

The benefactor bequeathed a bequest to the inheritor (or heir or recipient).

This is fairly formal. The way to say it normally is (after commenter user121863) is:

They left me an inheritance.

As to what to call the relationship between the two verbs 'inherit' and 'bequeath' they are not antonyms (they don't appear in thesauruses as such). They are called

inverses

of each other. An antonym or opposite of X is something that is -not- X, but for relations that are naming the same action but go in the opposite direction, one is the inverse of the other.

'Emigrate' and 'immigrate' aren't inverses but are obviously on the opposite sides of -something-. It's all about where the speaker is in relation to the person who moved countries. Someone emigrates from here to there, but they immigrate to here from there. Some kind of inverse relation could be extracted from these, but on the surface the words are not antonyms or inverses.