score:8
The train journey to Bremen would have been:
First leg: Zagreb (Agram) - Zidani Most (Steinbrück)
This is on the so-called "Croatian railway". The 80 km would have taken 2 hours.
Second leg: Zidani Most (Steinbrück) - Vienna
The 360 km on the Südbahn would have taken 12 hours, including traversal of the Semmering pass at 898 m above sea level.
Third leg: Vienna - Magdeburg
This answer shows a direct connection from Vienna to Magdeburg taking 21 hours. If that did no longer exist in 1895, changing trains in Dresden would not have lost much time.
It is also possible there would have been a mandatory stop at a "Control and Registration Station" run by the Norddeutscher Lloyd. They give "mainly sanitary reasons" for this, but also describe typical visa check procedures:
Einem vorwiegend sanitären Zweck dienen die Kontroll- und Registrierstationen an den Grenzen und im Inland. Die Kontrollstationen entstanden ursprünglich an der preußisch-russischen Grenze bei Ausbruch der Cholera und sind seitdem im gemeinsamen Betrieb des Norddeutschen Lloyd und der Hamburg-Amerika-Linie beibehalten worden. In den Kontrollstationen werden die Einwanderer einer ärztlichen Untersuchung unterzogen, müssen baden und ihr Gepäck desinfizieren lassen. Außerdem dient die Kontrollstation dazu, die über die Stationen nach Deutschland hineinfahrenden Passagiere daraufhin zu prüfen, ob sie als Auswanderer nach überseeischen Ländern, als Erntearbeiter für Deutschland oder als Einwanderer nach Deutschland zu betrachten sind.
A station on the route at Leipzig was opened in 1904. Whether there were comparable controls at the border in Bodenbach before that I cannot say.
Fourth leg: Magdeburg - Bremen
The so-called America line was taken by a lot of travellers from south-eastern Europe to Bremerhaven. Its 290 km to Bremen would have taken 6 hours. Since 1894, there was a special train by the Norddeutscher Lloyd from Berlin-Lehrter Bahnhof to Bremen, which would have passed through Magdeburg, although Lloyd admits that only a minority of its passengers were using this service:
Bei diesem Zuge handelt es sich im wesentlichen um solche Passagiere, welche erst im letzten Augenblick von Berlin abzufahren wünschen. Seine Benutzung ist daher bis jetzt auch noch eine verhältnismässig bescheidene...
Fifth leg: Bremen - Nordenham
Between 1890 an 1897, passenger ships crossing the Atlantic could not be accommodated at the Bremerhaven Lloyd Hall and were sent to Nordenham across the river. I have no confirmed travel times for that part, but between one and two hours are realistic. Ship passengers of the Norddeutscher Lloyd would buy their transatlantic ticket at least in Bremen, and it would include a chartered train from Bremen to the kay and separate transportation of baggage directly from hotels in Bremen to the ship. Timetables for that train would depend on the departure times of the ships:
Der Fahrpreis für die Beförderung von Bremen nach Bremerhaven ist im Passagepreis einbegriffen. Die Abfahrt der Züge richtet sich nach der von der Flut abhängigen Abfahrtzeit der Dampfer.
Realistically, if you had started in Zagreb in the morning, you would have reached Vienna in the evening. You could have taken an overnight train to Leipzig. If you had to pass the registration there, it would have lost you that much time that at best you could have reached Magdeburg in the evening, maybe only the following day. That would mean you could have been in Nordenham on the third afternoon or evening at best. Whether you could embark on a ship the same day, or if you had to stay another night in Bremen, would mostly depend on the tides.
I initially was wondering if Bremen is a realistic port. As you can see, it takes some time to get there across the continent. I think it would have been possible to get to Trieste by train within one day, and the Austro-Americana had around 10 regular (freight) ships running to US ports in the Carolinas. They only started passenger services in 1905, but would it have been possible to book a passage on one of those? The question seems to be moot since you have other evidence for the use of the route via Bremen.
For a detailed discussion of emigration from Austria to America, see Monika Korntheuer: Der lange Weg nach Ellis Island - Emigration aus dem österreichischen Teil der Habsburgermonarchie über deutsche Häfen nach den USA um 1900. From the contents list, it seems to discuss travel organisation in detail.