[bandito2]

Yes that is a fuel evaporation canister. Hooks up to the fuel tank and to an evap purge control valve and eventually makes its way via small hose to air box or to the carb. Part of emissions control if the bike is equipped with that. But some apparently don't and some may have been removed by owners. One nipple on the canister is to ambient open air and the other nipple is a drain. If there were any hoses, they would be just short pieces sticking out that hooked up to nothing. As rks said, it is not likely the cause of your problem.

From your description, it seems more likely an electrical problem or need for valve adjustment. Could be fuel related; fuel pump problem or tank not venting causing vacuum that holds back fuel. Dying battery could indicate a charging system problem. If it is not charging properly, the engine electrical may be running solely off the battery. When there is not enough "juice" left in the battery, the engine will die for lack of electric to be able to fire the spark plug. Or there may be an errant grounding happening somewhere that might be running the battery down.

You've got some diagnosing work to do to isolate the problems then solve them. Approach it methodically to help avoid wild goose chases.
Start with the simple and likely causes. See if it will continue running if the fuel cap is loose to ensure that the tank is venting. If it runs well with it loose, but eventually dies with it closed, then you'll know it is not venting properly. If it dies no matter what, then it is likely not a venting problem. It differs for different bikes so I don't know what the procedure for your bike is to check for proper fuel pump operation issues. After sorting out any fuel delivery issues, you could start checking on the electrical since that may be an additional contributing problem. Will the battery take an hold a charge? Yes- then there still may be a problem with the bike's charging system. No? then it may have a bad battery. You may need a multi meter to help determine electrical issues.

That is just a start. I'm pretty sure there are trouble shooting tips gathered somewhere here. There are those here that may be better able to direct you. But my point is to go about it methodically; that should save you lots of grief and wasted time. And it is also a reminder too, if and when making changes of whatever type to your bike that they should be only one at a time. That way if any thing starts screwing up, then you know you'll probably only need to check the last thing you fidgeted with. Changing too many things without making sure they behave as they should can mislead you with symptoms that may appear to be coming from something else.