On the one hand you don't need to run a new circuit to replace a receptacle, on the other hand putting in a new receptacle alone does absolutely nothing to modernize your system. Millions of houses have old wiring and they are fine. The only good reason to just replace the receptacle is because it is falling apart or won't hold the plugs in anymore. The problem is all new receptacles are grounded and your wiring was done before a third ground wire was required. You are likely to either have BX spiral metal jacketed cable or knob and tube for old wire. You can tell if you look in your basement or inspect where the wire enters the inside of the box. It is most likely BX if the house was built after about 1910. if you have BX you can replace the receptacle with a grounded one that uses the metal box and metal jacket as the path to ground. You can either use a metal clip that slides onto the box or you can scrape the metal clean an put a ground screw into the box. To that you attach a bare ground wire that you connect to the ground lug on the receptacle. The existing black wire goes to the gold terminal and the white wire goes to the silver terminal. It isn't difficult, should take you less than an hour.
As to the insurance fears and all that, this is not really likely to be what's potentially dangerous about your home wiring. The main danger of old wiring is in the ceiling boxes where light fixtures are mounted. Over years and years the heat from those old hot incandescent light bulbs makes the insulation on the wires brittle, creating the potential for these slow short circuits that heat up in the box but don't quite blow the breaker or fuse. You are best advised not to mess with old fixtures. Doing things like adding a switch leg is ill advised. If you have to deal with it you can only hope that an enlightened electrician from the days of yore left you some slack. Then you can cut back the ends and get to good insulation. But that's another story.
You should be fine. As a homeowner you should develop a plan for upgrading your house over time. Avoid half measures. When you decide to rewire a section of the house disconnnect ALL the BX on a circuit and re-run it in romex.