Q: Does this ASI have a backup PSU e.g. standby battery in the event of electrical failure?
No, these do not have a backup battery.
Q: Are these Swift Airspeed Indicators FAA TSO Approved?
No, this is not FAA Approved and is intended for use in experimental aircraft only.
Q: Does this ASI need an electrical hook up to operate or just a pitot line?
Yes, this requires an electrical hook up. Per the tech data sheet this runs on operational voltage 9-30 VDC and a pitot line.