So skyward sword takes place before the establishment of the kingdom of Hyrule, Yet in the lanyard desert there are ruins with a hylian crest. How could this be?

I think it's the goddesses symbol here, not hylian crest. It's the only explain I can find

The goddesses crest looks similar but without the Triforce.

At this point I really don't know

That isn't the Hylian Crest. That's the Wingcrest, one of two symbols derived from the Goddess Crest, and it's important to note that it's located on top of the Temple of Time.

In the beginning, there was the Goddess Crest used during the Goddess Hylia's time.
The Wingcrest was later used to represent the Royal Family of Hyrule specifically.
The Hylian Crest is used to represent Hyrule and the entire Hylian race as a whole.

Quick reupload of comparison image because of silly morning typo.

The one in the OP doesn't look like any of those, looks more like a hybrid between the Hylian crest and the Wingcrest. It may be something different

>>7
The emblem on top of the Temple of Time is still referred to as the Wingcrest. The Wingcrest of Skyward Sword puts more emphasis on the loftwing part of the design.