Internal rate of return Trying to finding the internal rate of return R:

Net present value of costs is $\ Y(s)$

Net benefit is $\ Y(s+1)-Y(s)$ in every period t up to$\ \infty $


The answer is given as $\ R=\frac{Y(s+1)-Y(s)}{Y(s)} $
Is this the correct approach (NPV = 0)?:
$\ Y(s) = \sum_{t=1}^{\infty}\frac{Y(s+1)-Y(s)}{(1+R)^t} $
Assuming the present is taken to be period 0, your approach is correct, subject to one caveat.
As you suggest, the internal rate of return is the discount rate at which the discounted present values of all the costs and all the benefits of a project sum to zero.  In this case, you are given the stream of benefits, and the right hand side of your equation correctly represents the discounted present value of that stream.
So far as the left hand side is concerned, instead of being given a stream of costs to treat in the same manner, you are just given the net present value of costs.  If the question is from a textbook and that is all the information given, you can hardly do anything else than use that net present value.
The caveat is that it's unclear how that net present value of the costs has been obtained. If it was calculated using any discount rate other than the true internal rate of return, eg an organisation's standard discount rate for evaluating the net present value of proposed projects, then (except by coincidence) your approach will not give the correct internal rate of return (and the given answer will be wrong).