Painful? Depends on your perspective.

You may be familiar with the Orient Express, probably from the Agatha Christie novel, or one of its numerous adaptations. This was a direct train line from Paris to Istanbul, then known as Constantinople. It was certainly around back in 1912, and this is a 1902 timetable, running on the same line it would have been in 1912. As you can see, the train departs from Paris at 7AM, and is scheduled to arrive two days later, a bit before 11PM, at Constantinople.

However, our prospective noblewoman needs to get to Paris first. Later, there would be a direct line from London all the way to Istanbul, including a train ferry across the English Channel, but a 1912 traveler has no such option.

Fortunately, we have access to Bradshaw's Guide and its April 1910 timetables. It advertises several express services from London to Paris, including night services that could get a traveler to the city of lights at around six AM. Assuming everything went as planned, this would quite possibly leave them with time to spare to board the Express.

As for getting to London? I am not very familiar with the show, but in the fiction, the titular Downton seems to be located between the villages of Ripon and Thirsk. While Thirsk is the closer of the two, Ripon happens to be on a route that takes us directly to London. A morning line departs from the village around 7AM and arrives at King's Cross at around 1PM, leaving plenty of time to prepare for the overnight passage.

So, how long would it take? Three days at the absolute minimum, closer to four considering how late the train arrives to Constantinople, and that's at a pretty rapid schedule. Timetables may slip, trains may break down, and any number of other mishaps can delay a traveler.

How much would it all cost? I found a possible price for the London-Istanbul leg of the trip in 1914 at 22 pounds and 11 shillings, citing "Le voyage à Constantinople: l'Orient-express," a source I unfortunately cannot definitively verify due to my lack of French knowledge. However, a quick machine-assisted translation makes me think it's credible enough for this answer. Adjusted for inflation, the figure comes out to being in the vicinity of 2800 modern pounds. Including the cost of the Yorkshire section and the return trip, the total price would probably come out to about 6000 pounds.

Does three to five days of rather expensive, non-stop travel, as well as a return trip of the same length, sound like a painful journey to you? I think it's a fair assessment.