This is a list of Sri Lankan athletes completing in the 2012 Olympic Games, the sports they're playing in and when you can watch. Where you can watch is given at the end, but note that there's no guarantee that these events will be shown.
July 29th (Finished)
Badminton: Thilini Jayasinghe
Sri Lankan Thilini Jayasinghe is in Group M of women's singles badminton, against Ratchanok Intanon of Thailand and Telma Santos of Portugal. They play each other and the top player advances.
On July 29th Thilini lost to Intanon 21/13 and 21/5. Santos hasn't played yet.
Swimming: Heshan Unamboowe
Heshan Unamboowe finished second in his heat for the 100m backstroke, losing to Ally Bradley of Barbados. The next round was based on the top 16 and his time of 57.94 wasn't good enough. In the final, all the racers finished under 54 seconds.
For more on this day, check out The Papare.
July 30 (Badminton And Shooting)

Badminton: Niluka Karunaratne - 1:37 PM
Today, 27 year old Niluka Karunaratne of Galle is facing 8th seeded Kenichi Tago at 1:37 PM Sri Lankan time. They are the only players in Group C, so I think Karunaratne has to win this match to advance. I'm assuming that our TVs will be showing.

Badminton: Thilini Jayasinghe - 6:12 PM
At 6:12 PM Sri Lankan time, Thilini Jayasinghe (27, from Peradeniya) will face Portugal's Telma Santos in what I assume is an elimination round. She's already lost to Intanon of Thailand so she has to beat Santos pretty convincingly to have a shot.

Shooting: Mangala Samarakoon - 1:30 PM
31 year old Mangala Samarakoon of Nuwara Eliya is representing Sri Lanka in the 10m Air Rifle and 50m Rifle. Today he's competing in the qualifiers for the 10m Air Rifle at 1:30 PM Sri Lanka time. If he gets through the finals are at 4:45 PM the same day.
August 1 (Swimming)

Swimming: Reshika Udugampola - 2:30 PM
Reshika Udugampola of Colombo will be in lane 7 of the second heat of the women's 100m Freestyle. Her qualifying time of 1:04 is not insanely competitive compared to the 0:53s and 0:54s people are throwing down, but I hope she competes well. All of the heats start about 2:30 our time, I've put the general time cause the heats are short and you might as well watch them all. And this information is not insanely easy to parse.
August 3 (Shooting)
Shooting: Mangala Samarakoon - 1:30 PM
In our final event (AFAIK), Mangala Samarakoon is representing the country in the 50m Rifle Prone, ie, shooting while lying down. The qualification is at 1:30 PM Sri Lanka time. If he gets through, the finals are at 4:30 PM.