I have never seen Arima used with non-normal data (well it was always assumed it was normal). Arima assumes stationarity and I don't see how that data will be stationary. It may be that differencing will address the issue.

If you just want to predict you might see how good a job exponential smoothing does. It does not assume linearity nor (for point estimates anyhow) normality - or at least time series analysts don't pay much attention to the later condition and the first is certainly not assumed.

It will be infinitely easier to do than Arima with exponential data

. You can compare the results with a hold out data set and see which does better. For either method you need at a very minimum 50 data points.