Ako budemo završili projekt na vrijeme, priuštit ćemo si kavu u mirnom kafiću.

Because Croatian commonly uses Future II (also called “future perfect”) in the ako-clause to express an action that must be completed before the result happens.

• Ako budemo završili... = If/once we have finished... (completion first)
• ...priuštit ćemo... = ...we will treat ourselves... (result after)

Using Ako ćemo završiti... is generally not standard for this meaning.

It’s Future II:

• auxiliary biti in present: budemo (1st person plural)
• past participle (active): završili (masculine plural / mixed group default)

So: budemo + završili = we will have finished (in the condition).

Not necessarily. Završili is the default participle for a mixed group or when gender isn’t specified.

• all-female group: Ako budemo završile...
• mixed/unspecified: Ako budemo završili...

Priuštiti (si) means to afford (oneself) / to treat oneself (to something).
Priuštit ćemo is a common shortened spelling/pronunciation of priuštiti ćemo, but the standard written form is usually:

• priuštit ćemo (very common)
• also acceptable: priuštit ćemo si as in your sentence

The verb stem often drops the final -i before ćemo in natural usage.

In standard Croatian, the future auxiliary ću/ćeš/će/ćemo/ćete/će is typically written separately from the main verb:

• priuštit ćemo = we will treat ourselves
  This is the normal way to form the future with ću-type auxiliaries.

Si is a reflexive dative clitic meaning to ourselves. It marks that the benefit goes to the subject:

• priuštit ćemo kavu = we’ll afford a coffee (less “for ourselves” emphasis)
• priuštit ćemo si kavu = we’ll treat ourselves to a coffee

Because kavu is accusative singular, used for a direct object (what you will have/buy/drink):

• nominative: kava (coffee, as a subject)
• accusative: kavu (coffee, as an object)

U + location (where?) takes the locative case.

• kafić (dictionary form) → locative kafiću
• adjective agrees: mirni → locative masculine singular mirnom

So u mirnom kafiću = in a quiet café (location).

That would be accusative and would typically imply movement/destination (into a quiet café), not being located there:

• u mirnom kafiću = in (staying/being there)
• u mirni kafić = into/to (going there)

Sometimes, but it changes the nuance:

• Ako = if (condition, not guaranteed)
• Kad (often kad/ako) can mean when (more expected/assumed)

So Ako budemo završili... stresses uncertainty; Kad budemo završili... sounds more like you expect to finish and then you’ll do it.