The Crossman C11 BB airgun would be totally unsuitable for hunting as you don't get much meat on a sparrow. For hunting rabbits and other small game you really need a bare minimum of 10-12 ft lbs energy to be humane whilst hunting small game. I doubt if the Crossman will get you 1-2 ft lbs energy at around 10-15 metres. (assuming 50-80 BBs per 12gram CO2 cartridge). The general rule is that with a minimum of 10-12ft lbs energy in .22 pellet you need to compensate somewhat with better accuracy for these very low ballistic energies. The Crossman C11 BB airgun will have poor accuracy compared to something like the classic Webley Tempest .22 air pistol (3-4 ft lbs). The Webley Tempest .22 would be unacceptable for hunting.

A more suitable alternative would be a .22 Magnum version of the Cricket Hunter Pistol. This would give a vastly more powerful hunting alternative firepower.

http://www.crickett.com/crickett_hunterpistol.php

If you are going to stick with a cheap air rifle CO2 solution then perhaps the QB78 .22 pellet rifle with an Archer XP tune kit (takes the power up to around 12ft lbs). The QB78 air rifle is reasonably lightweight and quite accurate, but will only give you around 25-30 shots per pair of CO2 12gm cartridges.

http://www.archerairguns.com/ProductDetails.asp?ProductCode=IBQB78D

http://www.archerairguns.com/QB78-XP-Tune-Kit-p/xptunekit.htm

In the UK anything above 12 ft lbs energy requires a fire arms license, even for a pellet air gun.