1. Is it contagious to humans? "Most" bird sicknesses are not contagious to humans so I would really doubt that this one is. If really unsure contact your vet to find out the exact type of sinus infection.
2. Is it going to hurt him if he swims in the water? He wants to go swimming in his drink and at the moment its quite challenging to stop him cos he is very determined. My mum has gone shopping to get a drink container with a little lid on it. I dont know if I should just put plain water til she comes back; i dont wanhim to swim in antibiotics but if i give plain water he might fill up on that and not get any of the medicine! Playing in his water shouldn't be harmful as long as he is not in any drafts or very cool air. Make sure he is able to fully dry before lights out at night
3. Is it likely that he is going to recover? I am worried that he won't be able to recover cos budgies are so tiny and he is starting to get a bit older as well. Budgies recover from sicknesses quite well as long as they get care quickly and when needed. There is always a chance with any animal of problems but I don't see any more of a chance of serious problems with a budgie than with any other bird that receives good timely medical care.