How do you keep a bee alive and what food can you give it?

It all depends on the condition in which you found the bee:

During the daytime in warm and/or dry weather:

If you see a bee on the ground and apparently unable to fly, it is probably dying anyway. However if you want to give it something in the hope of reviving it, give it sugar syrup made with five grams of sugar dissolved in five ml of water. Always use refined white sugar, not unrefined or any form of brown sugar because these can do more harm than good. Offer the drink to the bee on a spoon and with a bit of luck you may see it drinking. If it was just tired it will fly away when it feels better.

During the daytime in cold or wet weather:

The best place for a bee in cold weather is in the hive. However if unseasonably cool weather has caught the bee population off guard they will find it difficult to fly. If possible, place the bee in a safe place where the bee can draw warmth from sunlight. If the bee has gotten wet from rain it will not be able to fly again until dry. Again, place the bee in a safe location where it can dry itself off and return to the hive once the rain has stopped.

During the night in cold weather:

You might find yourself the surprised caretaker of a bee if you have brought potted flowers in from the cold. If this is the case, attempt to catch the bee in a container that will allow for air to reach the bee. The bee will be attracted to a bright source of light. Turn as many lights off as you can, and try to use an energy saving bulb so as not to damage the bee from direct exposure to any hot bulbs. If you lack a container and lid with holes, use a porous gauze-like material or a scotch-brite pad to place over the container. Use tape for securing the gauze, but make sure none is exposed to the bee which might injure the bee. Place a few coins on top of the scotch-brite pad to ensure the bee won't escape.

Releasing the bee from captivity:

If you are releasing a bee which found its way into your home during cold weather or rainy conditions, be mindful to release the bee into good weather conditions so that it will have the chance to find its way home, i.e. not cold or about to rain (or raining within a few miles). Remove the lid while the bee is calm inside the container. A bee exhibiting a temper may not be so understanding if let loose when angry.