Can I buy a Pass for last-minute travel?
Yes, but it does depend how last-minute you plan to travel. Express shipping can be as quick as X days. If you don't plan on using trains that require reservations, you only need to order X days before your trip. If you need to make some reservations, you'll need X more days of lead time.
If you are buying very last minute, you can always have your Pass delivered to your first destination in Europe as long as someone, like a hotel receptionist, knows to expect it and is available to sign for the packet. Check the estimated delivery date in the checkout and use this as an indication of where its best to ship your Pass.