1 month trip including Lviv, Cracow, Budapest and ???

Everyone's travel style is different, everyone's interests are different, so this may or may not anything you would be interested in, but

First go to Netflix and watch Winter on Fire. If it moves you, Fly into Kyiv. Maidan looks new again, but the history is still there and very relevant. Kyiv its self can keep you busy for three full days with no trouble at all. One of my favorite cities. So lets call it four nights. I like the Senator Maidan. Great little hotel in a great location.

Then on to Odesa for 3 - 4 nights. The flight will cost about $80. The Duke Hotel behind the opera house is really nice and well located for walking around town.

Then on to Lviv for 6 nights. Two choices, the national carrier with a stop in Kyiv or an airline I have never heard of that has direct service. Price about $130 either way. Why so long? Get a guide and head south through the Carpathians and do an overnight stop before returning back to Lviv for one night prior to moving on to Krakow. The Rudolfo is a really nice well located hotel.

Then on to Krakow on the train. I understand its pretty much a full day trip, but the carriages are new and the route has been upgraded (at least partially) to high speed trains. I believe you will have to make at least one change.

Five nights in Krakow makes good sense.

Now, how are you going to get from Krakow to Eger? There are direct flights to Budapest for under $100. It would be a shame to fly into Budapest, just to get on a train. So, spend three nights in Budapest but rent the Airbnb (there are some good ones in town) for your entire stay in Hungary.

After 3 nights in Budapest, leave the heavy luggage in the apartment and hop the train to Eger for 2 nights, 3 if you are a big wine person. Senator Haus right in the center of everything.

Then return to Budapest for a few more nights. You can easily do 5 to 7 nights in Budapest and still want more.

With a little tweaking, that's a month.

Traveling from Lviv to Kraków you could stop along the way in the charming Polish town of Przemyśl. Really off the beaten path as far as international tourists are concerned, Przemyśl has a long and rich history due to its advantageous geographic location and is the second-oldest city in southern Poland (after Kraków).

Because of its layers of history, the town has plenty of architectural eye candy, especially in the baroque style old center. If you are interested in WW1 history, Przemyśl fortress was the site of longest siege of the First World War, which was a crushing defeat for Austrian troops (as Przemyśl was then part of the Austro-Hungarian Empire). One can still visit the impressive ruins of the fortress where the siege took place. There is also an adjacent military cemetery, in which victims of this battle (one of the biggest in WWI) are buried.

Among other historical sites there is also very nice Royal Castle, built by Casimir III the Great of Poland, in the 14th century. Przemyśl also has one of Poland's only surviving pre-WWII Jewish Synagogues (built in 1910). It was not destroyed by the Germans during WWII, as it was used as a stable by the German army during their occupation. The Synagogue's original art nouveau architecture is now preserved as a library.

Regarding the Przemsyl fortress: In the Vienna Army History Museum (Heeresgeschichtliches Museum) you can see an armoured cupola (gepanzerte Kuppel) with a shell hole of the fortress Przemsyl.

It was indeed horrific, "crushing" ... those early battles ie, Przemsyl, Lemberg, in August-Sept 1914 between the invading Russians and the Austrians resulting in what so aptly described by the US historian G.A. Craig "the virtual destruction during the battles around Lemberg (Lviv) of the flower of Austria's junior officer and non-commissioned officer corps, a blow from which the Imperial and Royal Army never recovered. "

Have u considered Prague. Lots go do a and very tourisst friendly. There are also day excursions to take as u have the time. Ie Cesky kumlov.

Like James I was going to suggest adding Kyiv in Ukraine and then, since you don't mention having been to Austria I would suggest Vienna. Start by flying to Kyiv then Lviv, Krakow, Eger, Budapest ending in and fly home from Vienna. Sounds like a great trip.