Plumbing is actually pretty easy, especially for water and wastewater, someone with basic DIY skills can do this with modern tools and equipment like PEX instead of the original copper.
The plumbing for propane gas is trickier, might be best to hire it out if you're not comfortable with it and it would require several hours which adds up to $$$ though parts are fairly inexpensive.
You can build out most of your trailer with parts you find at hardware/plumbing/big box stores. If you're experienced and comfortable with the cabinetry and eletrical, you'll be able to estimate your time and expenses on your own for that.
Right off, $5K is going to be too low. A minimum of $10K parts alone is probably more like it, and if you have to replace the axle and other running gear, it'll go well above that.
But I think you might be putting the cart before the horse-- first, it sounds to me like you need a basic RV primer on systems/usage. There is probably already one here somewhere in the Forums, but a quick search didn't turn one up, so I'll do my best to help.
There are two basic ways to use an RV:
1) Full Hookups. This would include being hooked up to some external "city water" supply via a garden hose style attachment that is supplied by the campground on a utility post alongside the parking pad, and sends water to all of the faucets and systems inside your RV via standard city water pressure, just like a standard home. A full hookup site would also include either a 30A or 50A 110VAC electrical supply to run all the AC systems on board, this connects to a cable from inside the RV that runs into the RV's breaker box. And finally, a full hookup site would include sewage hookups to evacuate both black water (your toilet water) and gray water (shower, sinks).
2) Partial or No Hookups. This would exclude one or all of the above. The absence of all hookups is often called "dry camping" and/or "boondocking" for various reasons that are probably obvious. This means you'll need to source everything from on-board your own RV, and you'll need to store all of your outputs/waste as well. A lot of campsites that offer 30A electrical and city water, but don't offer sewage hookups, so you'd need a black tank to store the toilet water, and probably a gray tank to store shower/sink waste. The next step of campsite usually omits the electrical as well as city water. So you'd need an on-board freshwater tank to store water for showers/cleaning dishes and the like (many people carry extra bottled water for drinking and don't drink from the RV faucets at all). And to have electricity within your RV, you'' either need to run a generator that plugs into the exterior 30A 100VAC electrical cable, or you'll need a 12V
battery that runs various 12VDC lights/appliances within the RV (this is standard, pretty much all RVs have 12VDC electrical systems that parallel the 110VAC systems.
So, your choices in renovation are going to be driven by how you desire to use your camper. If you plan on doing a lot of dry camping, you're going to need different systems than if you plan always to plug into a full hookup RV resort. Personally, I like to change things up so I need the versatility of having the option of doing completely dry camping, or having full hookups.
The basic systems are:
1) Freshwater. The freshwater system has two sources, as discussed above. Either a city water connection from the outside, or the freshwater tank which uses a pump to move water within the coach.
2) Waste Water. For modern RVs, the drain of the toilet goes into a blackwater tank which stores the waste until you're ready to dump. If you're not at a full hookup site with sewage, then most campgrounds and some truck stops have a dump station to dump your sewage.
For modern RVs, the drain of the shower and sinks goes into a gray tank, which is either plumbed directly to sewage hookup at a full hookup site, or is stored in the gray tank until you reach a dump station.
3) 110VAC Electrical. For smaller RVs this is typically 30A. For those giant bus-style RVs with 2 or 3 air conditioners and 20 flat panel plasma TVs, they use 50A. The 110VAC is used to run lights within your RV, and can also be used to operate a refrigerator, the water pump, the water heater, microwave, air conditioner, TVs, DVD players, and other various appliances.
4) 12VDC Electrical. This is tied to the battery, most modern RVs come with a 12V
deep cycle marine battery. When plugged into 110VAC, modern RVs use a converter/charger to run the electrical systems on-board and to charge up the battery. When not plugged into 110VAC, then the 12VDC battery runs various lights, fans, and other electrical items on the RV, including some RV refrigerators that are specially designed for DC operation.
5) Propane gas. Typically used to provide fuel for stove, oven, water heater, furnace, and RV-style refrigerators. Older vintage RVs also sometimes had a lantern-style light or two inside that run off the gas. Modern RVs usually have two propane tanks mounted to the tongue, these are most likely 20# or 30# tanks each. Airstreamers like to get two 30# aluminum tanks and polish them, it looks great.
6) Running gear. This includes the axle(s), springs/shocks/struts, brakes, wheels, tires, and anything else involved in the driving of the RV. On vintage RVs like yours, typically some or all of this needs to be replaced.
7) Structure. This is the frame (possibly, even likely rusted at this age), the subfloor (possibly, in fact very likely rotted at this age), and all of the various fasteners and such. This will need to be assessed and addressed ASAP to make sure your Airstream is structurally sound.
I think that's the basics. So as you can tell, how you want to use the RV will determine which systems you might want to replace, leave alone, eliminate completely (I tossed my furnace, no need for it down here in Texas).
Hope that helps, good luck!