How do I analyze and design only a part of the system?
This component is dependent to other parts of the system and also has many dependencies in the system. The customer is suggesting to me which part should I do first.
How can I ensure that the components of the system are correct?
How can I minimize change (say half of works done) due to later user stories?
Agile is not just about developing system in chunks. There are many practices that you need to apply in cooperation in order to be successful.
For example, you architecture must be built to embrace change and your code should be sufficiently covered with the unit-tests to support constant refactoring.