nothing depends on docker here, it is just a way to install nginx and oauth2-proxy that is somehow platform independent. But I’m not sure if you can have this kind of flexibility to install both on hieroku.
Hertzer have server auctions that starts at 28 usd / month without upfront cost and you get a okeish bare metal server that you can use for a lot of stuff. It is what we do for our internal apps.
If you want to stick with hieroku I would have a look at the cloudflare they have some confusing products that seems to be doing authentication of other apps, that seems to be replacing oauthproxy that they were building some time ago: GitHub - cloudflare/nginx-google-oauth: Lua module to add Google OAuth to nginx.
I say confusing as I’ve spend like 20 min trying to figure out which one you can use and I failed :), but it seems to have the auth you are looking for:
https://developers.cloudflare.com/cloudflare-one/identity/idp-integration/google
If you manage to make it work please share in this thread.