In order to avoid any possibility that EA would attempt to bill an expired card and thus possibly flag my EA Master Account as being problematic in some way (I don't know if they do this or not, but I want to avoid it if they do!), below are the steps I followed on the two occasions when I had new card information to put in. (The first time I had to do it was when my existing card "expired" and bank sent me a new one with same number but new expiration date. Second time was when the bank decided to send a ton of people new cards because their existing ones might have been compromised.)
1. On the Account Management site, log in and then cancel each and every active UO account.
2. Next, you're going to update your existing card's expiration date OR put in a new card's information. To do this, you'll need to select a currently cancelled account from the drop-down list near the top center of the Account Management site. You can do it with an account you don't intend to really reopen, if necessary, by just not completing the renewal process after you've put in the new/updated card info. If you don't have any currently cancelled accounts, you might have to wait until one of your still-active accounts hits the end of its current subscription period and shows up as cancelled.
3. With the cancelled UO account selected on the drop-down list, click on the button with that UO account name showing in the "Reactivate a Closed Game Account" section of the site.
4. Pick a billing cycle on the next screen and then click on the Continue button.
5. On the next screen, click on the "Manage Accounts" link in the upper right corner.
6. On the next screen, you'll see a brief description of your existing stored card information and a link in white to "Add a New Credit Card." For existing card(s), there will be an option to Edit the card's information, which is what you'll need if you want to change an existing card's expiration date.
7. Depending on what you need to do, either edit an existing card's info or add a new card. After you've done that, you should be able to hit the orange "Back" button near the bottom of the screen to go back to the page where you select your payment method.
8. Now either proceed to finish reactivating an account, or just click the Home link at the top if you were just using any old closed account to get to the "Manage Accounts" pages and then cancel out of the reactivation process.
9. Reactivate any other closed accounts as they run out. When you get to the page where you select your payment method, you should now see either your updated card's information or a new card's information, depending on what you did in the previous steps.
I hope this helps. Just remember to try to "cancel" your subscriptions before EA has a chance to try to bill a card that's either expired or has been cancelled because the bank sent you a new one!