When posting a new listing I get a message that "Field name" custom field is required, but the field itself is nowhere.
Actually I don't see custom fields at all.

https://shmonka.com/index.php?page=item&action=item_add

Sorry your page required login to publish so could not test.
Try to test with default theme to make sure it's not theme bug.

I have removed requirement for login - you can see now.
It does "starter"'s issue, as on "sigma" custom fields appear.

Can I expect a fix? Or this theme will not be updated?

Theme is out of support, it may take a while.

Just tested on demo and everything works OK.
https://starter.mb-themes.com/item/new

Ajax calls are disabled in item publish.php of theme as they are fully handled by Osclass core.

Could you please explain what should I do in order to make it work or me?

First make sure you use latest theme version.
Next make sure custom field is checked on categories you expect.
Then check browser dev console & network tab after selecting/changing category, where ajax call should happen and get custom fields with plugin data.