It depends on whether the bike is old or the repairman is old.

We often hyphenate modifying compund words to clarify the meaning:
   My grandfather is an old-furniture salesman.

Sometimes no clarification is needed:
   An antique dealer could also be called an old furniture salesman.

Modifying words preceding the noun are often hyphenated:
   My six-year-old son likes to play with toy cars.
   American-football players are usually men.

Modifying words following the noun are not hyphenated:
   My son is six years old.
   Some men are players of American football.

There are some exceptions, so check a dictionary:
   Time-sensitive documents are those which are time-sensitive.
   A left-handed catcher would catch the baseball left-handed.

Compound adjectives that are comparative or superlative are often hyphenated:
   That is the highest-priced car I have ever seen.
   I would need a longer-term loan to purchase that vehicle.

But, of course, there are exceptions:
   He is the most talented youngster I have seen this year.

An adverb (-ly) compounded with other modifiers is not hyphenated:
   This is a highly rated bank.
   They sell publicly held securities.

The three things in the title are examples of how we form compound words.

new moon is the open form
   post office, real estate, half sister, attorney general, grade point average

firefly is the closed form
   secondhand, softball, keyboard, notebook, butterfly, workmanship

six-year-old is the hyphenated form
   daughter-in-law, over-the-counter, mass-produced, one-eighth, twenty-three

Which form should you use? Check an authoritative dictionary.

To further confuse the issue, the following are all corrrect:
   My six-year-old son.
   My son is six years old.
   He is a six-year-old.