If you are aiming to increase the properties of any polymer, your first and foremost task should be to achieve a sufficient molecular weight and low PDI values (optional). The property like strength of a polymer increases with the increase molecular weight up to a certain point and then saturates.
Other ways or methods to increase the property of PS is to incorporate nano-fillers into your PS to have a nano-composite. For example:
Nano-clay can increase the barrier properties and strength.
Graphite can increase the strength manifolds.
Aluminum Nitride can provide the excellent dielectric properties.
Boron Nitride can provide high thermal conductivity without affecting the electrical resistance.
Silicates in particular pattern can decrease the flammability.
You can blend it with high strength polymers. You can incorporate fibers to make composites out of it. You have 1000's of combinations and many ways to achieve, what you are looking for.
The strength of polystyrene can be increased by various methods including by increasing the molecular weight and crystallinity and also by introducing long chain branching. By the addition of appropriately surafce modifed carbon nanotubes at small weight% the tensile strength of polystyrene can be increased.
There are many ways by which polystyrene can be strengthened. One of the easiest methods is to copolymerize styrene with "very little" amount of divinylbenzene. This will introduce inter-chain crosslinking through sigma bonds at a number of sites.
When the end product is obtained, you will observe similar properties to those of polystyrene except for strength (which will be enhanced).