I did a stretch of commercial fitout work that was basically this, so here's what actually made the difference for me. First, the tool matters more than technique. If you're on a big hammer drill, get your hands on a proper SDS rotary hammer, the lighter class around two or three kilos. For small anchor holes they punch through concrete way faster, so each hole spends less time under load, and the drill itself is a lot easier to hold over your head. If you're already on an SDS, drop down a size. It's the seconds of resistance per hole that wreck your shoulders, not the total hole count.
Technique wise, stop death-gripping it. Loose hand, let the weight travel down through your arm bone, and stay directly under the drill instead of reaching off to one side, which is what ussually happens as the day wears on. Bracing it against your shoulder is a bad habit for exactly the reason you think, so replace it with something under the drill. Even a cut broom handle with a little cradle on top takes most of the weight off your arms. Better still, some guys build or buy a wheeled base with a vertical post and a drill mount, so you roll it along the floor and wind it up to height, kind of like the drywall lift idea mentioned above but purpose built. Crude but it turns an all-day arm workout into a light one. Also, that trigger finger aching at night isn't normal muscle tiredness, that's a nerve or tendon complaining. Rotate onto other tasks when you can, and if it keeps up past this job, get it looked at before it becomes permanent. And have a quiet word with whoever's running the job. Thousands of overhead holes is exactly the kind of task where they should be supplying better gear or splitting it across the crew, not watching one bloke grind himself into an injury because nobody asked.