is it true that if a spacecraft travled into space from Earth, it will ultimately reach the other side of the Earth (supposing it travled though the universe)?
It might eventually if our universe is topologically closed. Consider an analogy: The Earth's surface is a 2 dimensional topologically closed surface. If you set out in a particular direction and travel far enough, eventually you'll go right round the Earth and end up back where you started. Similarly, it is possible that our universe is a 3 dimension topologically closed space. If that is true, then if you travel out into space in a particular direction, eventually you will "wrap around" and end up back where you started. Recent astronomical observations seems to suggest, however, that our universe is not actually closed.
Notice by the way that it would be incorrect to think that you would arrive at the "other side of the earth". I feel sure that the time required to "circumnavigate the universe" (assuming such a thing is possible), the earth would have rotated enough that you could might well arrive at any point!