Will PostgreSQL server work if I install the service in mounted partition
Yes, you can move your base directory to any location you wish where the postgres service has write access and enough storage available. There are a few things to note

your startup script needs to set it's PGDATA directory correctly to the new location
your clients that connect locally have to set their PGDATA to the new location.
Tablespaces in postgres add a lot to complexity. Better is to try to prevent them being used at all.

Tablespaces are directories that are pointed to by symbolic links in $PGDATA so you can keep them if you like. They will be resolved.