koreader is an android app, for e-book readers based on android.
from what I've read it can be installed on the kobo (may need root)
here's a picture of how to set up koreader
https://wotaku.wiki/guides/manga/opds
the only thing to mind is that the opds address is not the calibre web one but <calibre address>/opds if I'm not wrong.
Also my calibre web does have a download button for the ebooks so if there is a browser on the kobo you should be able to download them. Are you using "calibre web" or "calibre" web?