You can use faker to generate a random word and then just add that word to the end of your email with a + it would look something like this
cy.get('someElement').type(`something+${faker.random.words(1)}@example.com`);
Then you can just navigate to your own email address and click the link from there. The only issue you are going to have is that you wont be able to do this all is the same test case as cypress does not support cross browser testing.
Also Im not sure if the activation link redirects you or not, but if it does you wont be able to navigate to that link either.
Here is an awesome article showing how you can use MailTrap to do this:
https://humble.dev/testing-an-email-workflow-from-end-to-end-with-cypress