Unfortunately, the chances of your getting this operational are small. I say that because of the 'Down Conv Reqd' sticker. This means that the unit is built to operate with a down converter that brings the GPS signal down to a lower frequency (like an IF - Intermediate Frequency) so it can be sent over a longer distance. There would be the GPS antenna, the down converter, the long cable, and then the XL-DC. The GPS receiver in this unit is incapable of receiving GPS signals directly - it has to have the down converter to operate. Now it IS possible to DIY this, but that's going to depend on your skill set, and exactly which board is in the XL-DC. I outlined some of this process in my message: <https://www.eevblog.com/forum/projects/symmetricom-truetime-xl-dc-gpsdo/msg885491/#msg885491> - more in that thread. If I recall, the most common IF was 34.075 MHz, so if you're comfortable building up a low phase noise LO and DBM, you may just get it working. It will still have the WNRO problem though.
The battery may well have been replaced, but these are long service life batteries, so who knows

You can get replacements if you need to.
Although old, I never had a power supply problem in mine. Of course, due to age I suppose it could fail at any time, but I wouldn't worry about it too much.