I drove over to Reims just 10 days ago. I don't think they have Uber. Parking can be an issue. For a taxi you probably want to speak some French and use a local company.
Try downloading the Moovit app. They cover Reims.
The tours can be spendy, depending on the house. And they are pretty much sold out for the next month already. I just went to a couple of the retailers in town and bought a case of Champagne from small, less well known, houses at Vinocave, and some 40 year old Armagnac at Vintage. You might also consider looking at the local supermarkets. I got two nice bottles of Mumms at the Carrefour in Varennes for 20% less than they would cost in Reims itself.