Is is possible for an isosceles triangle to be an acute triangle?

Yes, but a triangle is acute only if all three of its angles are acute (< 90o).
An isosceles triangle can be an acute triangle, but it doesn't have to be.

An acute triangle can be an isosceles triangle, but it doesn't have to be.
That depends on the sizes of its angle because an isosceles triangle has 2 equal sides and 2 equal angles which could be 70 degrees, 70 degrees and 40 degrees which are all acute angles that add up to 180 degrees.