I know you don’t want a car, but it’s going to be tough to work construction without one. Carpooling with co-workers would be the only other option that could potentially save you money. Plenty of my Mexican co-workers did that when I worked road construction.
If I were the foreman, I’m afraid I would never hire an employee that didn’t have 100% reliable transportation, every day of the week, without question. I’d need all contractors to have the ability to bring tools to the job site, bring their own hard hat, vest, tool belt, grab materials on the way to work if needed, run over and check on another job site, the ability to run into town, lumber yard, hardware store, etc as needed.
These may not all apply to your specific job, but the #1 demand out of a construction employee is reliability, and I would never trust someone without their own transportation.
The solution isn’t all that bad though. Pick up a used car for a few grand and never worry again. In my opinion, the stress of having to find transportation every day would far outweigh the stress of owning a car.