My general debugging philosphy: use a scope to ensure signal integrity. When the analogue signals are known to be within the spec for digital signals, switch to debugging digital signals in the digital domain. If protocol analysis is required (e.g. decoding UART/SPI etc) then consider something like a Bus Pirate or printf statements.
There are significant differences between the 2465 and 2465B, both in terms of capabilities (autoscale and measurements) and some repairs (A5 SMD caps, calibration memory) that might be needed. There are also many commonalities (mains filter caps, ICs). All are well-described on this forum and elsewhere. (Std caution: when operating out of the case, have gentle cool air blowing across the ICs on the main board).
Replacing through hole X/Y mains filter caps is easy, and should be done before they do damage.
Replacing A5 caps (n/a 2465) requires basic SMD skills and equipment. It might be necessary to replace a few standard SMD components nearby if the caps have damaged them.
Replacing the ICs is trivial, many are available on fleabay from sphere, qservice and elsewhere. A few are scarcer, but still available.
My main concern would be the calibration memory; if it has already failed then you will have to completely recalibrate the scope. If it hasn't failed then you could copy the contents (via the screen if necessary!) and insert them into the new memory. My attitude is to make a video of the contents as seen on the CRT, and only when necessary will I replace the components.
I have no knowledge of the 2440 and Fluke, but I seriously dislike the early small portable DSOs I've played with.
If you can see the 2465 showing a good display (preferably with the correct risetime!) and the service screen showing that all tests are passed, then I would consider that option. But would that offer you usefully more capability than your 2440?