There's a mission to assassinate an NPC for 160k. That's a lot of money. I wonder if that much money means I need a better ship and weapons than the initial ones to be able to accomplish this task.
Absolutely not. Think of the reward as the level of difficulty. If 160k is a lot to you, it's probably too difficult. The AI on those kinds of criminals is far better than the ones you find in warzones or RES nodes. And as a wise man once pointed out, your basic weapons are essentially laser pointers. You'll need strong beams (to knock out shields) and cannons (for hull damage) to do reliable damage.
Try some training in RES nodes, buy yourself a better ship (a Viper is a good choice for a combat-focused build), trick it out with full upgrades (don't forget shield cells), and then give it a shot. It'll cost you a couple million credits, but you'll stand a chance. Not a great chance, but by the time you're ready to hunt down one of those bad boys, you should be skilled enough to take him out. Just target the Power Plant, and it'll go nuclear once you knock it out.
Alternatively, just play it safe and wait until you can grab a Vulture or other higher-end combat-focused ship.