Salzburg to Hallstatt

I did. It takes 2 hours with one transfer in Attnang-Puchheim. Some connections take 2 hours 25 minutes.

There is a rail connection to Hallstatt. Attnang-Puchheim is on the main rail line between Salzburg and Vienna. At Attnang-Puchheim there is a spur line that runs via Bad Ischl to Obertraun. The stop just before Obertraun is Hallstatt Bahnhof, across the lake from Hallstatt. There is a boat that takes people across the lake from Hallstatt Bahnhof to Hallstatt. You can find rail schedules on the Bahn website or the Austrian Rail website.

There are also buses from Salzburg Hbf to Bad Ischl through the Salzkammergut, but you will only find them on the Austrian Rail website. The buses take about the same time to Bad Ischl as by train, but you get to see some of the Salzkammergut; at least one of the routes goes along Wolfgangsee. From Bad Ischl you can take the train on to Hallstatt Bahnhof or continue by bus. The bus station for Hallstatt is in Lahn, Austria, about a km south of Hallstatt, near the foot of the funicular to the salt mine. There are a pretty good selection of accommodations in Lahn if you can't find anything you like in Hallstatt. The car parks for Hallstatt are in Lahn, and there is a closed pedestrian road along the lake from there to Halstatt.

It was easy to take the train from Salzburg to Hallstatt. When you exit the train, the conductor points out the short path to the lake where you'll see the boat en route to pick you up. A few minute ride across the lake takes you to the exquisitely charming town of Hallstatt.

I’d rent a car for the day so you can maximize your time. Maybe you can stop at Gosausee afterwards, which is amazing.

Does the boat in Halstatt run at the end of December /beginning of January?

Yes
http://www.hallstattschifffahrt.at/en/timetable/timetable.html

Does the boat in Halstatt run at the end of December /beginning of January?

The boat schedule follows the train schedule. For every train arriving/departing there is a ferry to the town.

Sould it be very, very, very cold at the end of the year, i.e. the lake is frozen, I don't know how this is handled then. :-)