Do you capitalize Twenty and Century in Twenty-first century?

No—you generally do not capitalize “twenty-first century.”

Use lowercase in normal sentences:

We are living in the twenty-first century.

Capitalize only in these cases:

At the beginning of a sentence

Twenty-first century technology is evolving rapidly.

In titles or headings (title case)

The Impact of the Twenty-First Century on Architecture

So, in standard usage: twenty-first century (lowercase) is correct.