So that's great now you've got the screen working perfectly!
One thing you might want to do is to change the 56 tactile switches so that it feels responsive as new, about 30-45 depending on currency, for example
VintagePlanet has them, selling on ebay in some country or direct order, very reliable and fast shipping! I just did that to my pawnshop jx-305, it's not long, just make sure you write it down the where and how to reassemble, and pay attention to the cables they have no plugging sockets and can get damaged easily I guess. You can also swap the pots and the encoder for new ones but it's hardly that they would become dead as tactile switches can be (or having you forcing on the buttons, which lead to break them and use the synth less and less as it's a pain to have it functioning well)
While you're at it you can wash the casing and the buttons with some detergent.
Then the next move could be to get to the attic and find your card reader that could handle 5V Smartmedia cards, because you will absolutely want to get hold of one of these cards to deal with easy backups and memory shortage, and you can buy one from this seller on ebay who is inexpensive, fast and reliable :
http://www.ebay.com/itm/2MB-5V-5volt-SmartMedia-Card-SM-GENUINE-Brand-NEW-Made-in-Japan-ROLAND-BOSS-/370652355092?pt=Digital_Camera_Memory_Cards&hash=item564c9a1a14 It's only 2MB cards he's selling, but it's plenty enough for storing and moving data around!
After that, about this jx-305 synth, I find personally it's a really greatly velocity scaled keyboard for playing synths parts fast and expressively (saying that from a pianist perspective), and it's got assignable pedal input for any cc you'd like, potentiometers and buttons for real time midi control, direct panel access to midi channels 1-7 and 10, a neat sequencer allowing for serious sketches and real time 8x midi looping (up to 8000 events in an 8 parts x 32 bar pattern at a time, plus song mode) and a nice real time mute and mixing ability as well. There's also a useful complex arpeggio at your disposal, complex yet easy real time groove control and swing quantization. There's also this real time phrase thing that I didn't yet investigated much as it seems not to relay midi. I won't be using the local sounds much (no time for that, but it's certainly worth investigating on this too!), but I can't imagine a better midi keyboard with such a great sequencer for the ~250 price range I paid for it (all expanses included). If you are tired of mousing around to make music, this machine can do a lot to start making music again for real!
So and about the latest OS 1.07, still no a good soul to pass it over please? I can't find it anywhere..