I have used the cable car, so I can only comment on the last part of your trip.

For the average tourist, not used to hiking in the mountains, the tour is quite strenuous. It's a 20min walk (90m elevation gain) from parking to the cable car, another 15min (70m) walk from the cable car to the entrance of the cave, and about 700 stairs (134m) to climb inside the cave. Most of the path outside is wide enough for a car to drive on.

It's 1120m elevation gain* from Werfen (or Tenneck) train station to the entrance of the cave. Given your hiking experience, this should be doable.

Be prepared for the change in pace and temperature, though. When you get there, you might be soaked with sweat, and about to start a slow guided tour at freezing temperatures.

Eisriesenwelt says:

Hikers

Marked hiking trails to the Eisriesenwelt are available from both Werfen and Tenneck. The ascent from the valley to the aerial cable car will take about 1 1/2 or 2 hours. If you also intend to hike the cable car route (exposed mountain trail - only for people with experience!), you should plan for additional ascent time of about 1 1/2 hours.

Note: The footpath from the cable car’s valley station to the summit station is a secured Alpine climbing trail in an exposed location. You will need to be free of vertigo and have good hiking equipment! Please contact us if you have any specific requests or questions!

Link: https://www.eisriesenwelt.at/en

*Werfen train station at 523m (Google Maps route to the valley station), entrance at 1641m (Eisriesenwelt)

Edit: I've been there again. Here are some pictures taken from the cable car: