B767 Captain reported multiple failures of unreliable airspeed; altitude; over-speed and loss of autopilot at FL390.

Narrative

We were assigned a flight to ZZZ. On the return leg to ZZZ1 we experienced multiple failures of unreliable airspeed; altitude; overspeed and loss of autopilot at FL390. I was the PM on this flight; Aircraft X. I was in the process of preparing a sandwich from our catering and still occupying the left seat when we got a brief overspeed warning. We were cruising at M.81 and if I had to guess we had at least a 15-20kt buffer to the overspeed cue when this situation began. The flight was operating in and out of occasional light chop with small airspeed changes so this caught me off guard as we had not even come close to the overspeed cue throughout the flight. The First Officer (FO) reduced thrust reacting to the warning horn and EICAS message of overspeed. The airspeed returned to normal for about 5 seconds (referencing the Captain (CA) instruments here and I don't believe the FO's instruments ever indicated an overspeed). Next up the airspeed shot up well into the overspeed range and the altimeter I believe started showing a descent but was departing FL390 which was our assigned altitude. The autopilot kicked off sounding the warning horn and multiple EICAS messages from unreliable airspeed; unreliable altitude; ground proximity system; aileron lockout and rudder ratio from what I can recall from memory]. I instructed the FO to pitch for 4 degrees ANU while I ran the unreliable airspeed memory items and set the power. I advised ATC we were having an issue and wanted to depart FL390 immediately for more dense air while we troubleshoot and that we were going to lose RVSM capability. I didn't want to be at FL390 trying to figure out what airspeed was accurate knowing there are charts in the QRH to fine tune this issue and we weren't close to the part of the checklist yet. I also did not want to create more problems than we needed such as a risk of stall or large airspeed loss being that high up in the atmosphere not knowing what was telling the truth to us. They cleared us to FL350 initially and then on down to FL270. We determined that the FO's instruments seemed to be indicating correctly and were close to the STBY instrument readings. However; I asked for a few altitude checks in the descent from ATC and we were off by as much as 1800 feet between the FO's altimeter and the reading from ATC (The transponder had been switched to the right side and the altimeters were all correctly set at 29.92). This led to more confusion for us. We eventually descended through the last layer of IMC and had VFR conditions and could see ZZZ2 in the distance. I asked for direct ZZZ1 and eventually finished the airspeed unreliable checklist and did a quick look over of the other issues such as rudder ratio and aileron lockout so we had an idea of what to expect. Eventually after multiple altitude checks the FO's altimeter and ATC's radar began to show correct readings matching one another. I [requested priority handling] with ZZZ Center and we continued to ZZZ1 due to how close we were and the very strong tailwind blowing us in that direction. I contacted dispatch via XXX.XX and advised our status and the system failures. We executed an ILS in VFR to Runway XX with the reduced flaps of 20 degrees to a normal landing. The FO did an excellent job hand flying this airplane essentially with raw data throughout this [priority handling event].