You may be able to find a "cheap micrometer" at a lumber yard or hardware store for under $40 US that would be good enough for your purposes. Micrometers, even in pawn shops etc. in my area, usually run more than $100 because they need to be "certifiable" for use in the industrial places where people here need them.
The Lee Valley micrometer linked by Cluin looks like an adquate one for your purposes, and at $13.50 is quite affordable. The price isn't consistent with assuming it meets typical "industrial grade" requirements in my local industry, but that's impossible to evaluate without putting it through the labs. For occasional use it should be quite suitable. Don't expect to be able to make accurate measurements to the 0.0001 accuracy cited, even if the micrometer is capable of it. That requires considerable practice.
The plastic verniers I've seen are generally not good for better than +/- .01 inch, (often +/- 1/128") and are pretty marginal for accurate measurement of string diameters, although they'll let you tell which string is larger than which other one. You can "eyeball measure" that accurately with a simple scale/ruler if you can find a decent one with 0.01" ticks and your eyesight is very good.
Your local hardware/electrical house may have a "wire gage" that looks like a flat circle with notches around the edge. One of these should be less than $10, and can be quite accurate for telling whether a wire is exactly a given gage, but useless if the wire is "something in-between."
Perhaps the "cheapest" way to make reasonably accurate measurements is with a "flat feeler gage" set from your auto parts store. You can probably get a fairly decent set for $10 or $15, possibly less. You need to produce a "crack" the size of the wire, which can be done with reasonable accuracy by screwing the jaws of a parallel jaw wrench (commonly called a "Crescent wrench" in the US) snugly down on the wire. Then select "feeler blades" from the set until they just fill the crack, as close as possible to the wire. Add up the thicknesses marked on the gages to find the wire diameter.
The accuracy here depends largely on how carefully it's done. The feeler blades should allow you to stack up any reasonable thickness to +/- 0.001 inch, but your overall stack height should be considered about +/- 0.005 or so if you use 3 or 4 blades, depending on your skill and patience.
There are numerous other "make-do" methods, some of which can be quite accurate. Even with the $100+ micrometer, "amateurs" often get false readings because it's quite easy to crush the wire when you screw the "mike" down on it and get a "smaller than actual" reading.
Almost any good music shop, if they do any repairs, should have appropriate measuring tools. You might be able to take samples in and sweet-talk them into measuring them for you. Getting it done for you by someone practiced at using the tools they have is probably the simplest and most accurate method.
And when you put new strings on, write down what size they were for next time.