Short translation

You won't find a designation for "last act" in an old play because the Latin designations written in plays are numbers, ie "Actus Primus" is conveying the idea of "Act 1" or "ACTUS I" if you like. (IE "Act No.1" rather than "first [opening] act").

In a play I found, of 4 acts, the first scene of each act is headed with the act and scene number written in words, and the second and subsequent scenes in each act are headed with the numbers in Roman numerals thus:-

ACTUS PRIMUS. SCENA PRIMA.
ACTUS I. SCENA II
[etc]

ACTUS SECUNDUS. SCENA PRIMA
ACTUS II. SCENA II.
[etc]

The last act is headed
ACTUS QUARTUS. SCENA PRIMA.

(THE ELDER BROTHER by Beaumont and Fletcher 1625 approx)

http://thunderbird.k12.ar.us/The%20Classics%20Library/British%20Literature/beaumont%20and%20fletcher/elder%20brother.txt

If you want a Latin phrase which means "last act", then "Actus Ultimus" would fit the bill, but you should be aware that this is not how an authentic period play would head its last act.

I suggest a translation for "the final act begins" might be "actus ultimus orsus est"... maybe someobody more knowledgable than me might confirm that?