In short, yes it took "that much time" (overnight) to go from Munich to Vienna by train. Bram Stoker's Notes for Dracula contains a handwritten timetable by Stoker, which has been called "accurate" in The Essential Dracula (by the scholars Raymond McNally and Radu Florescu).
For Stoker's timetable, see Pages 84 & 85 of the facsimile
As for the language question:
3 May. Bistritz.
Date and city of journal entry, as you surmised. Remember that the author can write about things taking place (i) on the same day as the journal entry, (ii) in the near past, (iii) in the distant past, and/or (iv) in the future. So, in short, it doesn't matter how many days before he is writing that the journey from Munich to Vienna took place. It could have taken place a month before or a year before. Here it just happens to take place two days before.
—Left Munich at 8:35 P.M., on 1st May, arriving at Vienna early next morning; should have arrived at 6:46, but train was an hour late. Buda-Pesth seems a wonderful place, from the glimpse which I got of it from the train
The actual journal entry. The author left Munich at 8:35 P.M., on the 1st of May. He arrived at Vienna "early next morning". This is telegraphed (short) speech for "early the next morning."
Which morning? The most natural way to interpret this is with reference to the departure time. The departure time was "at 8:35 P.M., on the 1st of May," so "early (the) next morning" is the morning of the 2nd of May.
Again he doesn't write about this trip until he has a chance to–and that is on the 3rd of May.
See also The new annotated Dracula, which should be of great help to you.
Mike Kozar adds
a modern train would take about 4 hours to make the trip from Munich to Vienna, but Dracula was set in 1897. Our hero seems to have expected closer to 9 hours, which would have meant an average of 47kph or about 30 miles per hour. That's consistent for a rural or poorly maintained rail line of the time...
Here is a nice blog of the trip. And here's another one.