It is possible to do an upgrade from Classic to Premium on the website. You can also upgrade WiFi. Login to your Celebrity account. Select Upcoming Cruise and then Manage Reservation. Scroll way down and select Drink Packages and you should see the packages. You should also see an upgrade option. If you don't see the upgrade, look for a show all packages option. Sorry, I have already upgraded using these steps so I can't give the exact final step.
You will not receive the additional Captains Club discount online if there is no sale. You do have to call for that, but they will not combine that with a sale price. However, Celebrity has been heavily discounting beverage and WiFi upgrades with their flash sales. Watch for Black Friday if you don't cruise before then. We received 50% off our beverage upgrade a few months ago. A few final things. Don't be afraid to book the upgrade if you see 30% or 35% off. That's better than Captains Club discount and if you find it lower at a later date just cancel and rebook at the lower offer. Lastly, this all applies to US residents only 95% of the time. Our UK, Australian and Canadian friends rarely see these sale offers - if that's you and you see a discount, grab it.