A solo run in Phantasy Star is (probably) not viable. The closest anyone could come, I would think, would be Myau.
At level 30, he has 214 HP and 56 MP. Assuming he could get to Lassic at full health (well, -2 MP for the TRAP spell), it would have to work something like this:
Myau uses HELP in his first turn. Lassic attacks for ~50 damage. 164/44
Myau attacks. Lassic attacks for ~50 damage. 114/44
Myau uses CURE for 60 HP. Lassic attacks for ~50. 124/38
Myau uses CURE for 60 HP. Lassic attacks for ~50. 134/32
Myau attacks. Lassic attacks. 84/32
Myau uses CURE. Lassic attacks. 94/26
Myau uses CURE. Lassic attacks. 104/20
Myau attacks. Lassic attacks. 54/20.
Myau attacks. Lassic attacks. 4/20
Myau attacks one more time and hopefully goes first.
That's five HELP assisted attacks. Lassic has 238 HP. Is it possible? I'm not really sure. Any healing beyond that wouldn't be helpful unless Lassic's attacks did significantly more damage (or less damage). I'm not sure how much variation there is when it comes to damage in that battle. Burgers would be useless.
Darkfalz has enough damage variation where Burgers would maybe possibly be effective there. Again, not really sure.
A solo run would be technically impossible in that game because the TRAP and OPEN spells must be used, but you could just hack the MRCL KEY into your inventory.