Learn Turkish: Kalkıp beni azarlamak cüretini gösterdi. - He had the audacity to go and scold me.

No, it is not literal here. In this lesson's context, kalkıp (from kalkmak, to get up) acts as a converb to express an unexpected, presumptuous, or audacious action. It adds the flavor of 'he went and did something' or 'he took it upon himself to do something' without actually referring to his physical posture.

Why does cüret take the suffixes -i-n-i to become cüretini?

It is doing two grammatical jobs. First, it combines with the infinitive azarlamak to form a noun compound, azarlamak cüreti (the audacity to scold), taking the possessive suffix -i. Second, it takes the accusative suffix -i (with the buffer -n-) because this entire compound is the specific direct object of the verb gösterdi (he showed).

Could I use tutup instead of kalkıp in this sentence?

Yes, tutup works very similarly to kalkıp for expressing unexpected or audacious actions. Saying Tutup beni azarlamak cüretini gösterdi would mean exactly the same thing: he surprisingly and boldly went out of his way to scold you.

Why use the full infinitive azarlamak rather than a personal verbal noun like azarlaması?