Yes, I would argue you are immune.

While this is not explicitly called out by the description of the staff, I think the text of the staff's effect and the spell lean heavily toward your strategy working.

As you have pointed out, the spirit in the staff is explicitly attempting to possess the user, and Protect from Evil and Good explicitly protects against possession. This seems quite reasonable even though the staff did not call out this spell specifically in its description.

As additional support, the staff's description says that Dispel Evil and Good can be used on the affected creature to exorcise the spirit and force it back into the staff. The effects of Dispel Evil and Good and Protection from Evil and Good have significant similarities, with Dispel being a higher level with some additional affects, and both share the line about ending possession. Therefore I would expect that if Dispel is meant to be able to use its effect to dispel the spirit, Protection should be able to use the similarly-worded effect to protect from it.

These two points together make me think that Protection from Evil and Good should be sufficient protection from the spirit of the staff attempting to possess you.

All creature types

A monster is defined as any creature that can be interacted with and potentially fought and killed.

The monster manual defines monster as the above. And just from this you can get your answer: this spell works on basically everything.

You attempt to beguile a creature that you can see within range.

The spell works on any type of creature assuming it meets the other requirements in the spell description. But there are no types excluded.

See this question for an in depth discussion on what the word creature means in game terms.

To quote from the answer in that thread though:

Creature is basically every living breathing (or undead) thing big enough to be considered at least CR0 (give or take).

So the spell works essentially on any thing adventurers might come across.