There are several ways the draining effects of magic can be represented in the system.
I'm currently playing a shaman in an Aether Sea campaign, and I think a thing or two can be taken from its magic system: if you don't roll well enough, you have to take Stress equal to the difference between the opposition of the spell and your total roll result. That would mean that so long as Drain isn't huge, it should recover after the end of the scene (instead of counting turns or the like). Overdraining would of course inflict Consequences.
Another alternative that produces a similar feel would be creating an additional stress track, and getting it filled under similar circumstances, but that seems like a step towards complicating rules that you want to get away from.
You might want to look into how Jadepunk handles Ally-type Assets. I think they provide a reasonable compromise between simplicity and customisation of Stunts representing ability to summon a helper (in this case, a spirit). Generally, the Asset system may be of use for building other shticks that are appropriate for Shadowrun.
You should charge Stunts for cyberware when it produces strong effects - effects that are approximately worth giving up one Refresh for (even if the Stunt is bought out of the allowance of the three free Stunts rather than by literally selling Refresh). Otherwise, use an Aspect. Any cyberware that is more conveniently handled as an occasional bonus (whether invoked or on which an Advantage is Created) or as an excuse to declare a Story Detail should be handled by an Aspect.
A Word on Stunts in General
Cyber-samurai aren't the only ones who might need a lot of Stunts if they have a lot of cool abilities. Casters may need some too if they can provide unusually helpful buffs or big damaging spells. So you may want to look at the System Toolkit for ideas on how to give players more Stunts - whether by increasing the number of Stunts given at the beginning, doubling Stunts purchased per Refresh, both, or some approach of your own. Finally, there's a radical option found in Tiānxià that allows buying certain Stunts with Skill Points (bottom of page 77).