Light aircraft pilot reported receiving a low altitude alert from ATC on approach to OLM airport when they descended below charted altitude in instrument conditions.

Narrative

I was on a planned IFR practice flight in IMC. I was in communication with ATC and receiving vectors to the Final Approach Fix. The aircraft was in clouds; rain; and light-moderate turbulence for the entire approach. The autopilot had indicated a temporary autopilot failure condition (possibly due to delayed initialization) earlier in the flight; so I decided to fly the aircraft without autopilot.After being given vectors to final and cleared for the RNAV (GPS) approach; I was told to contact the Tower. I was close to the Final Approach Fix and approximately at the minimum required altitude (2400 MSL) to cross it. The short flight; turbulent IMC; and temporary autopilot failure had prevented me from completing my destination briefing earlier; so I began looking up the Tower frequency while hand-flying the plane and updating the GPS unit to sequence to the Final Approach Fix.A minute or so later; ATC called me with an altitude alert. I realized I had descended approximately 1000 feet below the minimum crossing altitude as I crossed the fix. I let ATC know that I was okay and immediately slowed the descent and carefully gained some altitude back to intercept the glide slope. I emerged from the clouds and was able to confirm the airport in sight and switch to Tower.The factors contributing to the altitude discrepancy included the workload; weather and lighting conditions; and possibly not choosing a more appropriate scale for the GPS navigation map near the Final Approach Fix. My relatively low time in actual IMC may have also contributed in terms of stress.I believe the altitude discrepancy could have been mitigated by making flying the plane the first priority; navigation the second; and ATC communications the third. Engaging the autopilot later in the flight could also have reduced workload; as well as updating the GPS map scale for different phases of the flight. Completing a full destination briefing before leaving the ground and noting important frequencies would also have helped with such a short flight.