Only recent rail extensions I can think of were the Stirling to Alloa line which refurbished existing track for 5.6 miles and a new station at Alloa cost £85m. Borders Railway was longer, 35 miles, and although some of the old track bed was still there new track and bridges were reaquired at a cost of £350m.
Would think a completely new line even only 4 miles of track, a station and associated signalling would cost more than the cost of Stirling to Alloa line but less than the Borders. Cannot see Scottish Government forking out for a line to serve a football stadium.
Other option could be a light railway/tram line from Aberdeen to Westhill. Edinburgh trams network runs for 8 miles so a similar distance as Aberdeen Westhill. It cost £776m to build. Would assume a line in Aberdeen would cost similar so no chance Aberdeen Council will fund that. Then you would have similar legal wrangles on the routes as there was on the bypass route.