I was wondering if during the initial ship construction if players need to spend SP to purchase archeotech components (provided they have gotten access to it somehow). The backgrounds or complications don't state if SP must be spent. So is it free, costs SP or depend on the wording?

Planet-bound for Millennia background states that the ship gets a free Modified Drive and access to an additional archeotech component.

The Thulian Explorator Vessel background has access to and must purchase an archeotech component.

The Age of Rebirth option states that an archeotech component may be purchased.

The Reliquary of Mars complication says that an archeotech component must be selected.

And are there any other ways to get archeotech during the initial ship construction? Thanks.

You have to pay Power, Space, and Ship Points for these items. The Modified Drive for the planet bound vessel is Space cost only.

You can get one Archeaotech from your initial Warrant of Trade if you choose Age of Rebirth. This is the oldest one and gives you one extra Archeaotech piece. I have not found any others aside from the ones you listed.