The Java Runtime Environment (JRE) is still available for Ubuntu; it's provided by a third part repository with package name update-sun-jre. This obviously won't work for in-browser use when the browser has dropped the API it uses, but if your signing can be done offline, this will likely solve the problem for you.
Otherwise, you may wish to examine the reasons Mozilla gave for dropping NPAPI -- which all boil down to security. Flash and Java, the main users for that API, are both security nightmares, and the trend in browser design has been for several years to move away from those helpers to XML and HTML5 to do the same jobs. It may be a good plan to ask your bank when they'll be updating to more secure tools, or how you can assure the security of your connection to them if they insist on using outdated tools.
Another possibility, though I don't know how long it will remain viable, is to switch to the SeaMonkey browser. It's Firefox under the skin, but it looks like the old Netscape Internet Suite -- and, at present, still supports Java (and flash, via pepperflash-plugin). It has generally been a couple versions behind Firefox, so it may gain you some time for a better solution.
The last, and IMO worst option would be to use Wine to install and run Internet Explorer -- but I'd avoid even mentioning that this is possible when discussing this security issue with your bank.