Cat5e is fine, the difference between a straight-through and a crossover cable are only the order of the wires in the connectors at each end. Unless you are using devices that are a decade old they will have Auto-MDIX, which means that it doesn't matter if the cable is wired up as straight-through or crossover, the devices can adjust themselves as needed.
Cat5e will be perfectly fine for 1 Gb/s speeds, so that will be perfectly fine, the only thing to consider is shielded or unshield, but unless you plan to run the cable right next to a mains power line for a long distance then unshielded (UTP) cat5e is fine.