At least according to https://core.trac.www.ads-software.com/ticket/12702 and whatnot, custom post types don’t handle sticky posts at all, and the core devs have limited to no interest at the moment to change that.
That said, I’m sure there are many ways to mimic the idea, like a hidden custom field or a specific taxonomy term that can be used to query for a specific post, and output that at the top, in a “sticky-like” way. Just going to have to be a custom solution.