Is there a word for this emotion? Resentment over someone's good fortune without wanting it - Not quite jealousy/envy

I don’t think you will find a better word than begrudge for “regard as ill-deserved”.

I know I shouldn’t care, but I begrudge my ex his new partner a bit.

Merriam-Webster agrees that it needn’t have a covetous connotation:

be·grudge verb \bi-ˈgrəj, bē-\

: to think that someone does not deserve something

: to regard (something) as not being earned or deserved

Resent is another good choice.

I believe the best word to describe what you are feeling is indignation. Begrudge is a good word if you don’t care about the similarities with envy and resentment. According to Dictionary.com:

indignation (n): strong displeasure at something considered unjust, offensive, insulting, or base; righteous anger.

That is, indignation at the other person’s good fortune.

Incidentally, putting a few words together to make resentful indignation is part of the definition of another word: wrath. Again, according to Dictionary.com:

wrath (n): strong, stern, or fierce anger; deeply resentful indignation; ire.

Wrath is a really interesting word, but it is a very deep emotion. As food for thought, is this person really the object of your wrath rather than your indignation? :)

Begrudge and resent are good suggestions in a different answer, but I would like to add that if you act on the feelings you mention, you might be said to act out of spite. Also, you behaviour could be construed as spiteful.