Nernst equation and equilibrium
problem where one is supposed to calculate the concentration of Fe3+ ions after a solution containing Fe2+ was titrated using acidified permanganate, 


There are several things which may be problematic in your approach.


For the stated problem you do not need to invoke Nernst equation. Simply use the balanced equation. 
When potentiometric titration curve are to be estimated, it is far more convenient would use the half-cell instead of a full reaction. I am assuming the redox electrode is dipping in iron solution.


\begin{align*}
    E=E^{0} -\frac{0.0592}{5}\log{[Fe^{2+}]/[Fe^{3+}]}
\end{align*}

where $E^{0}$ is the iron half cell value. You should not calculate equilibrium constants from potentiometric titrations. 

See page 17 for a full example Potentometric Titration clculation


I do not know when people will stop teaching normality. It is obsolete (18th century concept). Your normality conversions are incorrect.



  The problem mentions the following concentrations for the titration:
  Fe2+:0.1 N (I assumed it was 0.1/5=0.02 M)
  H2SO4: 1N (probably 0.1M for H+)
  KMnO4: 1N (0.2M MnO4- following the same steps)


1 N sulfuric acid will be half the molarity =0.5 M

1 N permanganate will be 1/5th the molarity = 1/5 M

Check the concept of normality in your textbook.