Asol = NDSolve[{A'[r] + A[r] == 0, A[1] == 1}, A, {r, 1, 0}, WorkingPrecision -> 30]

{{A -> InterpolatingFunction[{{0, 1.00000000000000000000000000000}}, <>]}}.

NDSolve[{(B'[r] + Sqrt[A[r]] /. Asol) == 0, B[1] == 2}, B, {r, 0, 1}].

InterpolatingFunction::dmval: Input value {1.} lies outside the range
  of data in the interpolating function. Extrapolation will be used. >>

  {{B -> InterpolatingFunction[{{0., 1.}}, <>]}}