Is there a way to tell how old a battery is because nobody bothered to punch out the sticker of the month and year the battery was bought.

I do not see any other sticker indicating how old the battery is either.

 

Some batteries have manufacturing codes with dates stamped into the terminals or top surface. These may need to be deciphered.

Batteries in general last three years. A decent battery analyzer / tester will test the battery under load & give both a CCA output & lifespan/health.