Try minimizing the square of the difference.

alpha=.;t=.; (* remove any cached previous values *)
Quiet[
  NMinimize[{
    ((1-4 NIntegrate[t^(1-1/alpha)(1-t)^(1/alpha),{t,0,1/2}]/
    NIntegrate[t^(-1/alpha)(1-t)^(1/alpha),{t,0,1/2}])-0.2455)^2,1<alpha<5},
  alpha]]

which returns

{4.52028*^-18,{alpha->2.93201}}

and

alpha=.;t=.;
ListPlot[Table[{alpha,((1-4 NIntegrate[t^(1-1/alpha)(1-t)^(1/alpha),{t,0,1/2}]/
    NIntegrate[t^(-1/alpha)(1-t)^(1/alpha),{t,0,1/2}])-0.2455)^2},{alpha,1,5,1/8}]]

which will give you an idea how this behaves.

The reason for the Quiet is that it complains about complex numbers for some of your powers. Usually I don't like hiding warning messages when doing things like this, but this can at least show you a result.

Remember there are always at least six different ways of doing anything in Mathematica.

Try

Assuming[alpha>1,expr=1-2455/10^4==4Integrate[t^(1-1/alpha)(1-t)^(1/alpha),{t,0,1/2}]Integrate[t^-(1/alpha)(1-t)^(1/alpha),{t,0,1/2}]]
FindRoot[expr,{alpha,2}]

and

FindInstance[1-2455/10^4==4Integrate[t^(1-1/alpha)(1-t)^(1/alpha),{t,0,1/2}]Integrate[t^-(1/alpha)(1-t)^(1/alpha),{t,0,1/2}]&&alpha>2,alpha]

There should be ways to get NSolve and Solve to do this for you.

And there will be even more ways of doing this.

Thank you! You have helped me a lot. BTW, what do you mean by " there are always at least six different ways of doing anything in Mathematica." ?