What to do without copper?

Assuming you're not also unable to find coal to use as fuel in your ship, your best bet is to simply seek another planet in the hopes that it has some richer copper deposits, particularly if you've already done a pretty good job exploring and exploiting the resources of the starting planet. There's really no reason to limit yourself to your starting planet.

However, as of the latest patch, silver armor no longer requires copper armor to be created first. It does, however, require you to have upgraded your ship with the Starmap Upgrade Mk2. The penguin mothership (which drops the molten core, which you need to create the metalworking station, which you need to make the starmap) is pretty easy to kill now and I'm certain you could do it without copper armor so long as you bring along a few bandages instead. Once that's done, skip straight to Silver armor.

After you manage to get pixels to hunting bow, hunt animals and make leather armor set, it have same defenses as copper armor.

Use dirt blocks to make cover and hunt with bow from it.

If you have coal(in current version coal is pretty rare too), try travel to desert planet, fine sand is very easy to mine, and you can find copper ore often near surface of planet.