Similar question arose when @sanuroxz went to Amritsar.Where to get ready if we reach earlier than check-in.
He opted for Railway station waiting room.
But because you are going by road, you can goto Golden Temple Langarhall side Sarai. You can get fresh there in 5 star bathroom facility. For boys it is easy. Ladies can also go towards ladies bathroom and can get fresh.
However you can also have langar of Tea /Biscuits etc in morning just opposite to the old sarai, Have darshan in morning itself. Then proceed to wagah border after check-in.
If you want to understand location of the sarai, I can give it later.
Have rest at any of those sarai or even can sit at outer parikrama of Golden Temple Sarovar under shade.
You can deposit bags at any of Sarai's Locker Facility / Gathri Ghar.