To clarify starsector does NOT UPDATE.
What you will do is install starsector 0.9.1 into a new folder on your computer and then that program will work perfectly fine.
You'll also need new copies of all the ship and faction mods, as well as the three libraries (also nex). Mods like portrait packs or whatnot don't need to be updated.
All of this is actually GOOD, because if starsector did auto-update what would happen is ALL your mods would break every time that happened! Starsector has a very small filesize compared to most modern games so just installing a new copy is a totally acceptable way to handle things. If the entire game is a 200megabyte download then there's no reason not to just reinstall it.
edit: One reason you might be having problems with this mod (and not other mods) is that this mod has NO VERSION for 0.9. This mod was never updated to run on 0.9 after 0.9 was released. The author of this mod finished his overhaul of the mod very recently and it is made for 0.9.1. So old copies of this mod would work on 0.8.1 and new copies of this mod would work on 0.9.1.... there is no file anywhere that would be a copy of this mod which would work on 0.9. No one has ever run this mod with 0.9. So, there's nothing wrong with your files, you're just trying to do something that was never possible for anyone.
tl;dr
download the game here:
http://fractalsoftworks.com/2019/05/10/starsector-0-9-1a-release/then download all your mods over again just to make sure you have all the updates:
http://fractalsoftworks.com/forum/index.php?topic=177.0note: you should open the hyperlinks to mods which say they're updated for 0.9 (but not 0.9.1) because this post itself is outdated and some of the mods have very recently been updated to 0.9.1 and so this post doesn't list them as 0.9.1 also as I mentioned before things like the libraries are still working and required.