Are there any restaurants or places to grab a quick bite near the museum or park?
Not within the castle area per se but there are restaurants near the area with less than half-kilometer away which I think won't be a problem. Unlike the Fushima Inari Shrine or the Arashiyama Bamboo Forest that offers food stalls along the walkways, I think Iga Ueno did it intentionally not to have food stalls to preserve their long-treasured culture. Strolling for food stalls or restaurants within the area is also good because you'll appreciate the province/prefecture even more with its scenic views of the neighboring area.
Near the Uenoshi station there are some restaurants and cafes, as well as convenience store. If you don't have to have special meal, I think it would be convenient to walk a bit from the park to the station.
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