The broad question here is whether "non-recyclable" plastics can be turned into fuel. The short answer to that is yes. More on that below. However, the specific proposal to convert polystyrene to isooctane is not particularly feasible. I'll address that first.
The proposed process begins with depolymerization of polystyrene back to styrene. Although lab-scale processes have been reported that give as high as 80% yield of styrene monomer, there is not yet an economically feasible process for scale-up. In fact, if styrene could be depolymerized practically, it would not be considered "non-recyclable". Furthermore, if the styrene could be recovered, it would be much more valuable as styrene than as gasoline, so there would be no incentive to spend the money on the subsequent steps described by OP.
Nonetheless, if one wanted to make liquid fuel from styrene, there is no need to produce isooctane specifically. Liquid fuels are generally mixtures of a variety of hydrocarbons, and the exact chemical composition does not especially matter as long as the autocombustion temp, vapor pressure and other relevant properties fall within an acceptable range.
So rather than make isooctane, one could simply hydrogenate styrene to ethylbenzene, an already common anti-knock agent or all the way to ethylcyclohexane (a more likely final product than octene).
Going back to the original premise of getting liquid fuel from plastics, just as there is no need to produce a specific compound from the styrene, there is no need to recover styrene specifically from the polymer, nor even restrict one's self to a specific polymer type such as polystyrene. Instead, bulk plastic mixtures can be pyrolyzed to produce a mixture not all that different from some crude oils and gases. These liquids and gases can be used as fuels themselves or mixed with existing petroleum products and refined.
Many challenges remain to make such a process economically viable, however, not least of which is the cost of collecting and transporting plastic waste (generally quite low density) compared to recovering oil from the ground.
