What is my fastest option to get from Copenhagen airport to Tivoli by public transport on a weekday?
This is actually really easy - I've been to Tivoli twice, and it's conveniently right next to the central train station.
As Wikitravel says:
It takes twelve minutes by train to get from Kastrup to the central station (Hovedbanegården) in the city centre. You need a ticket for three zones which can be purchased from one of the automated vending machines or the ticket counter located inside the atrium and costs 36 Kr for a single journey. The Copenhagen Metro also connects Kastrup with central Copenhagen, with trains leaving every four minutes during the day and every fifteen minutes at night, taking fourteen minutes to the city center (for the same ticket and price of 36.00 Kr).
And then from the Hovedbanegården station, follow the signs across the street to Tivoli. Enjoy it!