This word is a more polite way to say old

You seem to be looking for a euphemism, csicska. «Old man» and «old woman» are plain terms that some people might prefer, but they are not euphemisms that are designed to take the sting out of plain language.

Barque’s suggestion makes sense to me. There’s no need to add anything to «gentleman», «lady», «man» or «woman» in a remark about somebody’s right to a seat. I hate euphemisms, but I don’t ordinarily go out of my way to offend others with harsh language either.

There you have it, csicska. There is no consensus in here on a polite term for «old». Some people suggest «elderly». Others don’t like it. You’ll have to choose something that satisfies you and rest content with that choice.

There are a lot of English-speakers in the world, and you’ll likely offend or annoy somebody no matter what you say. Try not to let that worry you too much. If you do, you’ll have to whisper everything you say or use another language for the rest of your life. The moral authority who first suggested «elderly» as a polite substitute for «old» was probably self-appointed anyhow.