If the apartment lease was signed and no one moved in or took posession of the apt can we cancel within 3 days if something came up?

You are thinking of a "3-day right of rescission" that is available in certain situations after signing a contract.  Unfortunately a lease is not one of them.

Frankly, you don't have many legal options. Having signed a lease, you may be held to its terms which includes paying rent for the entire term of the lease. The fact that you never took possession does not matter. 

That having been said, the landlord has a duty to "mitigate damages" (reduce) by re-renting the apartment and may not hold you liable under the lease once a new tenant starts renting. However, while the landlord must make a good faith effort to re-rent, that's no guarantee of success. You can help them and yourself by finding someone who will take over from you, either as a sublet or an assignment.  However, legally, you are responsible for the rent for the lease term or until the space is rented to someone else.