According to the 2002 CGEL, the usage of the expression "many a time" is fine and standard English.
But if you use a different count singular noun than "time" in that expression, such as in "many a person", then you might risk having that expression seen as being somewhat formal or archaic.
In the 2002 CGEL, page 394:
Many in combination with a
Many combines with a to form two kinds of complex determinative:
Many a is syntactically inert: nothing can intervene between many and a, and many cannot even be replaced in this position by its antonym few. Like a, many a always functions as determiner. It is found in proverbs such as There's many a slip twixt cup and lip, and in the frequency adjunct many a time, but is elsewhere somewhat formal or archaic. The many component indicates a large number, but the a has an individuating and distributive effect requiring a count singular head.
Great in a great many can be replaced by good, but one or other of these adjectives is required; for the rest, these expressions are syntactically comparable to a few. They function as determiner or fused determiner-head (simple or partitive).
NOTE: The 2002 CGEL is the 2002 reference grammar by Huddleston and Pullum (et al.), The Cambridge Grammar of the English Language.