Modeling multipath distortion and shifting by fractions of a sample
One approach is to upsample the signal, then delay it, then downsample (or, if feasible, you can do the rest of the processing at the higher sampling rate).

This general approach to modeling multipath is correct. However, you need to make sure that your model is realistic and/or useful for your purpose. A few things to consider:


If you add delayed/attenuated replicas of the signal to itself, then you're assuming you have line-of-sight between the transmit and receive antennas. The effect is different when you don't have line-of-sight.
The number of paths is important too. A channel with many paths is easier to understand theoretically, because you can make certain assumptions about the overall channel gain (e.g. that it's a Gaussian random variable).
The maximum delay is important too. A very large delay introduces very destructive effects on the received signal. Many, very small delays can be modeled as a single gain factor.
In a static environment, the channel is also static. If the antennas and/or obstacles in the environment are in movement, then the channel is dynamic, in the sense that the number of paths, and the gain of each, vary with time.