Off sick with stress for a month and possibly longer? Since when is stress a sickness?
Since a long time - it's a recognised condition. A rather serious one at that.
Just do it for the money you get paid. It only looks good on you anyway. The whole "well so and so's off i should get x and y " is kinda lame, just help out...good work ethic and all that.
Obvs if it goes on for a long time, that's different.
I find that attitude laudable. And most probably one I would follow.
However, it has to be fair - if the OP is going to be working many more hours, and if it does or may affect his work/life balance, he needs to be recompensed for it.
However, he also needs to show that he can step up when he needs to without being a princess about it. There could be others that would snap fingers off for a chance of a bit more responsibility or at least gain an understanding of what it is like to be the manager of that department/product.
Personally, I would see it as a L&D opportunity - as long as you're given the understanding/knowledge required to do the role.
What I'd recommend is, give it a go for a couple of weeks, see how you get on. Then, on your 1-2-1 or review (which I would hope would happen weekly in this situation) let them know how you're getting on. Be honest, if you're struggling, tell them, if it is affecting your personal life tell them. However, at the same time, dig deep and see how you get on - if you feel like you are being exploited long-term then of course do suggest some sort of compromise.
I wouldn't bring it up before even starting the role as it is all likely to be done unofficially. Undoubtedly if the directors/bosses are any good, they will acknowledge your efforts and offer some sort of discretionary reward. I know when I've stepped up as an interim manager in certain roles it went recognised and whilst I didn't stay in that role, I was given a nice extra bonus that year.
Don't be greedy, but at the same time don't be taken for a ride - it's a fine balance.
Ultimately you could potentially end up with the role if you do as good as or better job - but for that you need to grit your teeth and go for it, and not whine or worry about it. We've all done months of long hard stints of work in the past to get things done - it's not easy and you do have to make sacrifices, but you know what? It gets noticed and if anything helps to develop you as a person too.