Series prediction for any given time
Considering ocam's razor I would recommend to use the simplest model first and increase the complexity if the simple models fail:


Exponentially Weighted Moving Average, allows only for auto correlation with lag one
ARIMAX, allows for several lagged autocorrelation, seasonal adjustments and external regressor.
Fourier transformation, allows to fit more shapes time series but they are often more complex to explain to users


All these models can predict one step ahead and you can repeat this up to 60. Of course the confidence intervals become much wider these many steps ahead. Base R has many time series available. 

These two models work well for many time series data. I would only start to use neural nets when these are not working or if you are looking for say edge detection. 

Neural nets

If neural nets are the only option to go you can check this post or if you have matlab available this post. 

I recently found this very interesting article on medium that explains how to fit a Neural Net to time series data. A more detailed implementation guide can be found here. I have not tried either approaches but I thought that the may be useful for future reference.

Time series modus operandi

Note that in any case, before staring off modeling your time series it is highly recommended to take some preparatory steps:


Make your series stationary as explained here
Understand what elements you can decompose in your series as explained here


Hope this helps