I haven't heard this exact phrase before, but I live in the inaka. I'm assuming it's something like "yoisho" (however you spell it).

I can't think of anything that correlates with it in English. When an English speaker sits down, they might let out a sigh, or even say "Ahhh..." in an equivalent situation.

When lifting a heavy object, they might let out a breath then stop it halfway out as they tense up their muscles, making kind of a "Huっ" sound.

When my mother stands up (she's old with arthritis), she says "oo ouch oo ouch."

As for actual words, though, I don't think any exist.

Might I suggest 'there we go' as a natural sounding equivalent? Sometimes when older people talk to small children as they dress, feed, diaper, or buckle seat-belts, they might say when they have finished the task, 'there we go, now you're all set' or 'there you go, now you're all ready'. The phrase is kind of an interjection without much meaning, just an expression you might make after finishing a task that takes a little (or a lot) of effort. I can imagine someone leading a child or an elderly and weak person slowly up or down stairs, and stopping to encourage as they finish each step, 'there we go now, old chap!'For the case of an old woman, perhaps it is better to say something slightly different when sitting down or getting up. When sitting down something like 'oh my!there we go' or 'good heavens! there now, sittin' down' might do the trick. When getting up, something like 'alrighty, now up I go' or 'now then! let's get ourselves up' sounds appropriately elderly and old-fashioned, and expresses somewhat more articulately the grunts or little sighs of effort being made.