By train
Limburg has two train stations; "Limburg (Lahn)" for regional trains and "Limburg-Süd" for high-speed trains.
→ High-speed trains between Cologne and Limburg (Süd) and Frankfurt (Central Station and Airport) Limburg (Süd) won’t be available due to construction work from July 16 to August 12. ←
Coming from the north (via Cologne) most appropriate trains run through the Rhine valley via Bonn and Koblenz. Change in Koblenz to Lahnstein and take the train from Lahnstein to Limburg. It’s a beautiful trip but takes some time. As a kind of a replacement there are some Express buses between Cologne and Limburg.
Coming from Frankfurt airport, take a regional train at the Frankfurt airport "Frankfurt(M) Flughafen Regionalbahnhof" to Frankfurt central station and change to the regional trains (RB22 or RE20) to "Limburg (Lahn)". This train takes about 1 hr from Frankfurt central station to Limburg. From "Limburg (Lahn)" station it is a 10 min walk to the Dom Hotel. The entire trip takes about 1h 15min. (https://tinyurl.com/yfek8xcv)
Alternatively (which is also given by the Deutsche Bahn Travel planner), take the Bus X58 from Terminal 1 to Frankfurt-Höchst and change here to the regional train RB22 or RE20 to Limburg (Lahn).(https://tinyurl.com/4583et7a )
Coming from the south via Frankfurt central station, take the regional train to Limburg (Lahn) (RB22 or RE20). This train takes about 1 hr from Frankfurt central station to Limburg. From "Limburg (Lahn)" station it is a 10 min walk to the Dom Hotel. The entire trip takes about 1h 15min.
Exit the station, cross the bus terminal and enter into "Bahnhofstraße". Follow this street (which crosses a market place) and cross Grabenstraße. Then the hotel will be on the right hand side.