Simply collect and power-up every cyber elf without using any in normal mode, and beat the game once. After you continue from your old save, you'll get Jackson.
Mega Man Zero 1: Easy Mode
You can get a slightly easier game. If you finish the game and save after the credits, you can begin a new game with all of your old equipment, though they won't be leveled up yet. This means you can start with the upgrades from cyber elves in effect, like double life gauge and protection from spikes.
Mega Man Zero 1: Jackson Mode
It is possible to obtain this very rare cyber elf, but I didn't say it was easy. Play through Hard Mode and collect (as well as power up to the max) every cyber elf, as well as getting Ranking A or S on every mission. If you start a new game from that saved game, you'll begin with every cyber elf, including Jackson.
Mega Man Zero 1: Skip Credits
After completing the game and saving when the game prompts you to do so, hold A when the credits appear when you complete the game any other time. Note: You must have at least one save with a ''1'' or higher to the right of your total play time.
Mega Man Zero 1: Starting a game in Hard Mode
After completing the game, you can start a new game in Hard Mode by holding the left shoulder button down while starting a new file. In Hard Mode, weapons do not gain levels, and bosses have new attacks.
Mega Man Zero 1: Ultimate Mode
Once getting passed the challenge of getting Jackson, Ultimate Mode is a bit easier to unlock. Simply hold R when selecting a new game after playing through Jackson Mode and using every cyber elf to play Ultimate Mode. Though the name suggests otherwise, this mode allows you to start with all of your weapons maxed out and with the effects of cyber elves already in place; permanent cyber elves no longer effect ranking. Also, you can now perform any of Zero's charged attacks without actually charging them. To fire a semi-charged Z-Buster shot, press Down to Forward + B (doing this combination with the Z-Shield equipped will use a charged Z-Shield throw). To perform a charged Z-Buster shot, press Down to Forward, Down to Forward + B. To perform a charged Z-Sabre slash, press Forward to Down + B. To perform a charged Z-Rod swing, press Forward, Down, Down-Forward + B.