ご滞在 is a noun, and 予定 is also a noun. ＊「滞在する」is verb.　「ご」is affix for respect. So ご滞在の is simply modifying 予定。 You can re-write it as 「どのくらい滞在なさる予定ですか。」 （「なさる」is respective form of 「する」。）

I suppose you are having trouble in understanding the usage of particle (助詞-じょし) 'の' As far as my knowledge there are 3 ways in which 'の' can be used as below: A] Possessive Marker :- 'の' indicates ownership or attribution. It is similar to the English "apostrophe s ('s). " Ex:-1) これは私の本です。( Kore wa watashi no hon desu.)This is my book. 2)私の姉は東京に住んでいます。(Watashi no ane wa Tokyo ni sunde imasu.) My sister lives in Tokyo. The final noun can be omitted if it is clear to both speaker and listener. Ex: あれは私の(車)です。(Are wa watashi no (kuruma) desu.)That is mine (my car). B] Noun Modification: The noun before 'の'modifies the noun after 'の'. This usage is similar to the possessive, but it is seen more with compound nouns or noun phrases. Ex: 日本語の授業は面白いです。(Nihongo no jugyou wa omoshiroi desu.) The Japanese class is interesting. C] Apposition 'の' links the noun to the appositive that follows. Ex: 友達の恵子さんです。(Tomodachi no Keiko-san desu.) This is my friend, Keiko. I hope this could help.