Old electrical - changing out light fixture
Since the light works by the switch your new fixture won't need a switch (pull chain). I like pigtailing if the wires are connected through the fixture not sure from your question. Basicly the 2 wires that connect to the existing fixture connect go to the new fixture. If a metal fixture there should be a ground wire that connects to the ground in the junction box, if a plastic fixture there may not be a ground wire on the new fixture. You should not need to change the switch.