This works pretty well. Numeric ....

solpru = NDSolve[{D[y[x, t], t] == -D[D[y[x, t], x], x], 
    y[0, t] == Sin[0], y[Pi, t] == Sin[Pi], y[x, 0] == Sin[x]}, 
   y, {x, 0, Pi}, {t, 0, 1}, MaxSteps -> Infinity, PrecisionGoal -> 3,
    AccuracyGoal -> 8, 
   Method -> {"MethodOfLines", 
     "SpatialDiscretization" -> "FiniteElement"}];

    Plot3D[Evaluate[y[x, t] /. solpru], {x, 0, Pi}, {t, 0, 1}]

Analytical

sol2 = DSolve[{D[y[x, t], t] == -D[y[x, t], {x, 2}], 
y[0, t] == Sin[0], y[Pi, t] == Sin[Pi], y[x, 0] == Sin[x]}, 
y, {x, 0, Pi}, {t, 0, 1}]

{{y -> Function[{x, t}, E^t Sin[x]]}}

Hey. First ,I don't know the best Method options for this type,but this code below works:

ppR = 20;
solpru = NDSolve[{D[y[x, t], t] == -D[D[y[x, t], x], x], 
    y[0, t] == Sin[0], y[Pi, t] == Sin[Pi], y[x, 0] == Sin[x]}, 
   y, {x, 0, Pi}, {t, 0, 1}, MaxSteps -> Infinity, PrecisionGoal -> 1,
    AccuracyGoal -> 1, 
   Method -> {"MethodOfLines", 
     "SpatialDiscretization" -> {"TensorProductGrid", 
       "MinPoints" -> ppR, "MaxPoints" -> ppR, 
       "DifferenceOrder" -> 1}, 
     Method -> {"ExplicitEuler", "MaxDifferenceOrder" -> 4}}];
ys[x_, t_] := Evaluate[y[x, t] /. solpru];
Plot[{ys[x, 0], ys[x, 1/2], ys[x, 3/4], ys[x, 1]}, {x, 0, Pi}, 
 PlotRange -> All]

See another code:

ppR = 50;
solpru = NDSolve[{D[y[x, t], t] == -D[D[y[x, t], x], x], 
    y[0, t] == Sin[0], y[Pi, t] == Sin[Pi], y[x, 0] == Sin[x]}, 
   y, {x, 0, Pi}, {t, 0, 1}, MaxSteps -> Infinity, PrecisionGoal -> 1,
    AccuracyGoal -> 1, 
   Method -> {"MethodOfLines", 
     "SpatialDiscretization" -> {"TensorProductGrid", 
       "MinPoints" -> ppR, "MaxPoints" -> ppR, 
       "DifferenceOrder" -> 1}, 
     Method -> {"ImplicitRungeKutta", "DifferenceOrder" -> 2}}];
ys[x_, t_] := Evaluate[y[x, t] /. solpru];
Plot[{ys[x, 0], ys[x, 1/2], ys[x, 3/4], ys[x, 1]}, {x, 0, Pi}, 
 PlotRange -> All]