What time do air conditioned buses run from Colombo to Anuradhapura? When is the earliest bus leaving?
You have multiple options to link the two cities early in the day, including AC buses.
The buses from Colombo to Anuradhapura are frequent, but few are AC buses. The schedule from the Sri Lanka Transport Board between the two cities show buses leaving at 2.15 AM, 3.15 AM, 5.15 AM, 8.45 AM and then every 30-45 minutes until 9.30 PM. They all take 6 to 7 hours. This website does not show which are AC buses, but from the National Transport Commission website (which unfortunately has only dead links to timetables), we can see that rouote 15-1-1 is a luxury service (I suppose this means AC buses).
This luxury service runs every day and is the 2.15 AM bus. The cost is announced at 240 Rs. but since it supposedly applies to all these options, maybe (probably) the luxury service is more expensive.
As Rome2Rio shows, you can also take a train. You can search your own schedule but the first train leaving Colombo Fort to Anuradhapura (not Anuradhapura Town station, which is apparently not always served) leaves at 5.50 AM. It has only first class seats, available for 600 Rs. It arrives in Anuradhapura at 9.15 AM (duration of 3 1/2 hours).
The following train leaves from Colombo Fort at 7.15 AM, arriving at 11.15 AM and has 1st, 2nd and 3rd class for 600 Rs., 370 Rs. and 240 Rs. respectively.