wine-staging is not part of the repo itself, unfortunately (due to it using the same files as wine proper it was decided to keep it out, back in the day; I'd have to search the old dev tracker task to see what the exact arguments were back then).
Now, I'm sure there are some Wine managers out there that can download different runtimes (e.g. staging), but off the top of my head I can only think of lutris. While it is mostly used for games, I guess it would be possible to make it work for your use case too. (they simply call their Wine runtimes "Lutris-x.x", but they include the staging patches as far as I know).
But I'd wait for someone else to chime in who has experiences with wine-staging on Solus.
Last resort would be of course to simply compile it yourself.
