Will 1080p content on 4K display look as crisp
Obviously, it depends a lot on the quality of the displays. A high-quality 720p display may actually give better result than a bad 1080p one, and likewise a bad 4k display isn't that much use!

But assuming the quality is "the same" (though I'm not sure there is a good way to compare this objectively), 1080p footage will look at least as good on 4k display as it does on the 1080p one. In doubt, it can just upsample the footage, that's a lossless operation (in theory) and may in fact already improve the picture, since it's doesn't have the 1080p's pixel grainyness. Likely, it will also do some clever sharpening, which actually gives a "crisper" result, though that may well seem unnatural.

Remember that high-definition TV sets were on the market well before much HD material was actually widely obtainable, and even then it was a substantial quality improvement over the old standards.