Is it possible to make micelles to pick up Microplastics from water?
Lyophobic means solvent-hating, and here the solvent is water, so the lyophobic ends (presumably the tails, if we are discussing a normal detergent) will associate to form the hydrophobic core of the micelles. The headgroups meanwhile will preferentially interact with water, forming an interface between solvent and core. 

If you add detergent to a solution containing charged plastic particles, the detergent tail is unlikely to interact with the particle unless the particle also displays amphiphilic character. More importantly, headgroups with the same charge as the particle surface will discourage association with the particle. So your proposal lacks a viable aggregation mechanism for at least one reason (charged and hydrophobic regions are not likely to associate) and possibly for a second (oppositely charged regions of detergent and particle will repel). So you need to research carefully:


the expected charge of the particles
the choice of detergent, ideally with a headgroup with charge opposite that of the particle, and which resists precipitation due to the hard water but will encourage flocculation of the particles
scalability, which is a question regarding the volume of water you intend to treat


There is always the possibility that the solution contains ions resulting in charge inversion of the charged particle, another factor to consider.

You'd be well served to look further into the steps used to treat wastewater.