A319 pilot reported that departing as the sun was rising made seeing the dim CRT screens difficult and after an ADR fault appeared; the reporter questioned whether the ISIS speed indication was correct for takeoff.

Narrative

We arrived from ZZZ1 at XA:00 and arrived in ZZZ before the sun came up. After swapping aircraft with about 1 hour on the ground; we felt a bit rushed getting to the next flight to ZZZ2. As we left the gate the sun was coming up. As we departed [Runway] XXR the sun was shining in my eyes from the side and made it difficult to see all the screens properly even though they were all turned up to the highest. As we rolled down the runway since the aircraft had a light load it accelerated quicker than normal and I was late making the 80-kt. callout. V1 and Vr came very quickly and we were airborne in no time. Shortly after the after takeoff checklist was completed an ECAM NAV Air Data Reference (ADR) 3 FAULT came on; which was a repeat write-up. We completed the ECAM procedures which basically had us turn the ADR 3 OFF. Shortly after I noticed a red SPD label on the Integrated Standby Instrument System (ISIS) airspeed indicator. Since the conditions were VMC all the way to ZZZ2 and both primary airspeed indicators were working properly we decided to continue the flight. Afterwards I began doubting if we had proper ISIS speed indication during the takeoff. I can't say with certainty that I did. While enroute I first wrote up the ADR 3 and then the ISIS airspeed. The flight to ZZZ2 was normal and in VMC conditions.I believe having such an early wake-up call for an XA:00 departure from ZZZ1 to ZZZ just before operating flight to ZZZ2 had an impact on my alertness level. Also; departing with the sun rising from the left made it more difficult to read the Primary Flight Display (PFD); which then made it more challenging to read the ISIS. The fact that the aircraft was relatively light and it was cool; meant that the aircraft accelerated down the runway very quickly.