Learn Spanish (Latin America): Yo tuve tiempo para publicar la aplicación privada.

Tuve is the 'I' form of tener (to have) in the preterite tense, which we use for completed past actions. Tener is highly irregular in this tense, so its stem changes completely to tuv- and it uses a special set of endings.

Why do we use para instead of por here?

We use para when talking about a purpose or a goal. Here, having time had a specific purpose: publishing the app. A good rule of thumb is that if you can substitute 'in order to' in English, you'll almost always use para in Spanish.

Why is publicar in its dictionary form (infinitive)?

In Spanish, when a verb directly follows a preposition like para, it must stay in its infinitive form (publicar). Even if English might sometimes say 'time for publishing', Spanish sticks to the infinitive here.

Why does privada end in an '-a' and come after aplicación?

Aplicación is a feminine noun (almost all Spanish nouns ending in -ción are feminine). Because of this, the adjective describing it must also be in its feminine form, privada. Also, descriptive adjectives usually come right after the noun in Spanish.