How to close a popup alert which may appear any time while browsing the website in cypress?

To close whenever requires some sort of watcher, which doesn't quite fit the Cypress pattern.

Best approach would be to cy.stub() it so it never occurs. What type of popup do you have, what triggers it?

If it's timer based, you might be able to cy.clock() to freeze the clock. If you need to cy.tick(milliseconds) during the test, then follow it immediately with

if (Cypress.$('.popup').length) {  // non-failing check if popup is present
  cy.get('.popup a').click()
}