Learn Finnish: Koko seremonia alkoi itse paavin saapuessa. - The whole ceremony began when the pope himself arrived.

This is the core rule of the intensifying appositive pronoun. When itse is placed directly before a noun to emphasize 'the person or thing itself', it remains completely uninflected. It stays as itse regardless of what case the following noun takes.

Why is paavin in the genitive case here?

It serves as the subject of the temporal structure saapuessa (when arriving). In Finnish, when you use this second infinitive construction to say 'when X does something', the subject 'X' must be put into the genitive case.

What exactly is the form saapuessa?

It is the second infinitive of the verb saapua (to arrive) in the inessive case (-ssa). This forms a temporal construction (temporaalirakenne). The phrase itse paavin saapuessa is a compact, slightly more formal way of saying kun itse paavi saapui (when the pope himself arrived).

Why doesn't saapuessa have a possessive suffix at the end?

In a temporal construction, you only add a possessive suffix if the subject of the main clause is the exact same as the subject doing the action. Here, the main subject is koko seremonia (the whole ceremony) and the one arriving is paavin (the pope). Because they are different, you just use the genitive noun followed by the bare -ssa ending.