I just purchased a "leasing" license but I do not have time to activate it immediately. Will I lose days?

No, you will not lose days.

The rental period of your license begins to run when you set it in the software, not when you buy it. 
You can surely purchase a license for one month and not use it until several weeks after your purchase. The only date that counts is the activation date (see the question "How do I activate my software with the authorization code received by mail?").