A rebuilt engine should be just as reliable as a brand new engine would be.
When a vehicles engine is rebuilt entirely it means installing new parts such as bearings, oil pump, timing chain, head gaskets, valve seals, piston rings, pistons etc.
Basically the entire engine should be disassembled and have all the moving parts replaced with new ones with the engine rebuild kit.
You might also have a new camshaft and crankshaft installed but sometimes those parts are just turned but the crankshaft is usually replaced.
I rebuilt an engine in a truck and it's been going strong for 200,000 miles now and I would still trust the truck to go on a long road trip.
Rebuild the engine properly and it should last just as long if not more than a brand new engine would.
Take proper care and maintain the engine such as making sure you have enough engine coolant and make sure the oil levels are correct.
Change the oil every 3,000 to 5,000 miles and the engine should last a very long time as in you should get 200,000 or more miles out of a rebuilt engine before it needs rebuilt again.