I hold a garage on lease, will the Lands Authority give me permission to build over the roof of the garage?

No Lands Authority will not grant you such permission. This is because the airspace over the garage is not normally included in your lease. The granting of the airspace under any form of title is a disposal and will therefore have to be made in accordance with the Disposal of Government Land Act, which would normally mean that tenders would have to be issued.