There are many forms of travel into the city, a taxi will cost on average €30 to the Playa de Cibeles. If you opt for the cheaper option a train, from Terminal 4, which leaves every 30 minutes will cost you €5, and will take you to either Recoletos or Playa de Cibeles. There are 4 stops between the airport and Playa de Cibeles and the journey will take you just under 30 minutes.
Additionally, there is an Express 24-hour Shuttle Bus, which will transport you directly into the city for €10. This will take approximately 40 minutes.
A hop on, hop off Madrid City Open Tour Bus will take you around the city. It has multilingual audio commentary about the history of the buildings in 14 languages: Spanish, English, French, German, Italian, Portuguese, Dutch, Russian, Japanese, Chinese, Arabic, Catalonian, Euskera and Galician. One or 2 days passes can be purchased for €20 per person. You will also receive a useful map of Madrid and a discount voucher for various shops and restaurants, throughout the city, which may come in handy whatever your budget.
There are two different routes on the Open City Bus, you are can travel on, and you can board at Museo del Prado, which is the first stop of Route 1. However, you can hop on and off anywhere along the routes.