Immortal yes, goddess not necessarily, for reasons I just stated. Without further clarification or other evidence to show that she's a goddess, I wouldn't go around saying she is one.
Lord Tsukuyomi brought his relatives, including Eirin, to the moon. Eirin is older than Tsukuyomi. Tsukuyomi may be implied to be a god considering his implied sister Amaterasu is a god in Touhou, but that's stretching implications, and nowhere does it explicitly say Tsukuyomi is a god. Eirin also has heavy connection to Omoikane (a god) in nearly every way, and could be interpreted to be Omoikane, but again that's just guesswork. On the other hand, why even need to drink a Hourai Elixir if you were a god? You're already immortal. Lunarians however, are not immortal, and while on Earth continue their lifespan as normal. Therefore Eirin would have needed to drink the elixir to stay with Kaguya, unless Kaguya's power over eternity and her influence on Eientei already covered that. In which case, again, why even take the elixir?