No London City Pass includes free use of public transport. But you can order a Visitor Oyster Card from the Visit Britain Shop. This gives you convenient travel on the London Underground, buses, the Docklands Light Railway, river boats and local trains.
Instead of paying for each journey individually, the Oyster Card is topped up with credit. This is usually sufficient for a journey within the city centre (zones 1 + 2). You can set the amount of credit yourself (£10 - £50). When purchasing, you pay the selected amount plus a one-off activation fee.
A portion of your credit is deducted for each journey. However, there is a daily limit, which is usually reached after the third journey - after that you travel for free for the rest of the day. Once your credit has been used up, you can easily top it up at a ticket machine.
If you would like to order the Oyster Card by post, you should do this at least two weeks before your trip.