Congrats on your first post!
Thanks.
On the topic, the main issue for Krakow is basically on suppressing their own population's nationalistic desires. The Partitions and the Congress of Vienna were still a vivid memory by the time of the Krakow Uprising. And even before, the city was used as a smuggling hub for rebels in the November Uprising in (Russian-controlled) Congress Poland. For it to survive, its leaders would have to rein in on these sentiments. Too much, however, and the city would rebel against the government for being collaborationist, and that would churn out a pretext for any of the neighbouring powers to march in to 'restore order'.
On the other hand, a shrewd politician could try to play the partition powers against each other and keep the city a neutral state, but it's a very dangerous game, and frankly, there isn't much for Krakow to play to its advantage compared to its neighbours. Keeping it low on the international spotlight could help, but again, its very population is made up of people very eager to reverse the indignity of the Partitions, and ignoring any suppression of Poles in the partitioned territories is likely going to increase public discontent against their leaders, even start a rebellion outright.
Overall, I don't fancy their chances. Unless conflict hit any of the states, preferably at the same time as with WWI (with all the states preoccupied with each other and at risk of civil war), there's little way for Krakow to survive. In this case, I'd suggest Krakow either last an additional 60 years for that to happen (admittedly a
very long time to hold your anger for) or start throwing more butterflies to destabilize the other powers (e.g. Revolutions of 1848 degenerating into a free-for-all civil war all over Central Europe, and/or the 1905 Russian Revolution resulting in an early Russian Civil War). Krakow could even take advantage of the chaos and absorb Polish lands from any rebel-held domains, but it's a heavy gamble and could backfire hideously if any nation got a grip on itself and suppressed the rebellions.