Can I play the same unit I sacrifice to Heedless Resurrection?

No. You cannot kill a unit with Heedless Resurrection's cost and then resurrect that same unit.

When you play Heedless Resurrection, you first choose a target unit from your trash (Step 2: Make Choices). The unit you intend to kill is still on the board at that point — it's not in your trash yet, so it can't be chosen as a target. Choices are locked in during Step 2, and only in Step 4 do you pay the cost by killing a friendly unit. By then, your target is already set, so you cannot retroactively pick the newly-killed unit (FAQ #9523; Rules 355.5, 355.9.a, 355.15).