I haven't tried updating 'resolution' itself but here has been my general approach:
Assuming it's a field: https://docs.atlassian.com/jira/REST/latest/#idp1378416 That's what I used for my downloaded version of JIRA and my assumption would be that the on demand api would be very likely the same. What I did to be able to see all available options is to first make a call to https://docs.atlassian.com/jira/REST/latest/#idp1448896 and then grabbed all possible field options.
If the resolutions don't appear from the editmeta call it could be counted as a 'transition'. If so, then you'll want to use this instead: https://docs.atlassian.com/jira/REST/latest/#idp1351200 This has both a GET and a POST so you can verify you are doing it correct, but I'm pretty sure only a status counts as a transition (although when you are doing a transition you can also set the field options so yu can theoretically update resolution through there)
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