Skin, probably. Everyone forgets that your skin is an organ.

According to Wikipedia:

In a multicellular organism, an organ is a collection of tissues joined in a structural unit to serve a common function.

Following that, the oldest organ that forms, is actually the trophoblast on day 4-5, which will give rise to all non-embryonic parts.

But let's ignore that and only look at organs that one generally has with them for life, and only once the organ starts to actually form together as separate from pre-organ and 'cognate organ' cells. In other words, once the organs form as standalones.

Correct me if I'm wrong, but that would then make the heart, as it forms by Carnegie stage 9, 25 days after fertilisation. This is before the skin proper is developed.