I am looking for a budget option of getting to Anuradhapura straight from the Colombo international airport. I haven't purchased a plane ticket yet but I am planning to time my arrival before late afternoon.
If you're not willing to fly, Rome2Rio says that you can take a bus to Colombo fort and a train all the way to Anuradhapura, for a total travel time of approximately 6h10:
Other options include taking a bus to Pettah CTB, another bus to Puttaiam and a taxi from there.
There are 3 budget options to you that come to my mind. Note that Colombo International Airport is about a 30 minute drive from Colombo if you take the expressway.
1) Bus to Colombo (Route Num 187) and then train to Anuradhapura.
This is the more travellesque option, but it does take some time. If you plan to come by late afternoon, there is a train at 3.55pm from Colombo fort station. Or else take the night mail train, but it will be early morning the next day when you get to Anuradhapura. If you take the night mail, make sure you book tickets beforehand through a travel agent. Here's the timetable.
2) Catch a bus without going to Colombo
From the airport, take a taxi to the Chilaw-Colombo main road. There are buses from this route that travel to Anuradhapura via Puttalam, and they are available throughout the day. This is probably the quickest and cheapest option. You can try any bus with route number 87 (Colombo - Jaffna/Vavuniya/Anuradhapura) and also other buses. Just ask a local. The downside is that you might not be able to catch a seat.
3) Take a bus from Colombo Fort
Time consuming and exhausting, not recommended because of the slum atmosphere around Colombo Fort Bus Stand, but the upside is there are frequent buses departing to Anuradhapura and since they start here, you'll definitely catch a seat. (Route Numbers 87, 15, don't take 57 it takes more time.)
There are of course other diverse budget options, depending on the time and your willingness to 'experience'. Let me know if you need to know them. Hope this helps.