Is it normal that an ELT activates during a strong up draft in flight?
The manual for your ELT doesn't say exactly what triggers it (or I overlooked it) and the EUROCAE ED 62 standard is behind a paywall. The FAA ELT standard TSO C-126 isn't on the FAA site, possibly because it seems to be based on a commercial standard RTCA DO-204A, which is also behind a paywall.
However, your ELT complies with C-126 and I found a thesis called The sensitivity of aircraft emergency locator transmitters to nondistress impact events about false activation rates for ELTs that includes this (p.27):

TSO-C126 ELTs follow the specifications of RTCA DO-204, Minimum
Operational Performance Standards for 406 MHz Emergency Locator
Transmitters [15]. In these documents, the RTCA defined a crash as
when the longitudinal axis of the aircraft experienced a velocity
change of -3.5±0.5 feet per second (fps), with a minimum of
deceleration of 2±0.3 Gs.

If the turbulence you encountered exceeded those tolerances then I assume it would set off the ELT. I don't know if your unit needs repairs or a new battery, you would need to check with your repair shop. In the US there are required inspections and rules on when to replace batteries (14 CFR 91.207) but if your aircraft isn't a US one then that wouldn't apply to you.
I would get someone to look at your back before you worry about the ELT, though :-)