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:
[66]
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.
2
Many a time = on many different occasions/often. It is a somewhat "folksy" manner. https://books.google.com/ngrams/graph?content=(many+a+time)&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2C%28many%20a%20time%29%3B%2Cc0 and https://books.google.com/books?id=sIBBAQAAIAAJ&pg=PA33&dq=%22many+a+time%22&hl=en&sa=X&ei=A4cDVbTcJo7msATqo4K4AQ&ved=0CDsQ6AEwBg#v=onepage&q=%22many%20a%20time%22&f=false
– Tᴚoɯɐuo – 2015-03-14T00:54:58.297reminds me of a family anecdote - my aunt, when she was young [1950's] on arriving home late one night [probably 'late' == after 10pm in those days] was asked by her mother, "Young lady, have you seen that clock?" - to which she replied, "Yes, many a time." – gone fishin' again. – 2015-03-14T01:07:12.430
1To my native ear, "many a time" doesn't seem like something that you would normally say (and doesn't sound like it's correct grammar, even though it might be correct). I've never heard anyone say it – Jojodmo – 2015-03-14T01:33:17.907
1@Jojodmo - it's perfectly good English, just antiquated. It's actually quite common in Northern BrE - as are many other antiquated forms. – gone fishin' again. – 2015-03-14T11:58:20.750