The Advanced Photon Source plans to upgrade to a multibend achromat (MBA) lattice that will dramatically decrease the electron beam emittance, thereby enhancing the x-ray brightness by two to three orders of magnitude. Electron beam focusing in the MBA requires small-aperture vacuum components that must also have a small impedance so as to minimize rf-heating and collective instabilities. As part of this effort, this paper focuses on coupling impedance measurements and analysis of certain critical Advanced Photon Source Upgrade vacuum components. Impedance measurements of accelerator components have traditionally been done with the coaxial wire method, which is based on the fact that the Transverse ElectroMagnetic (TEM) mode of the coaxial cable can mimic the Coulomb field of a particle beam; however this measurement technique has various limitations. This paper describes our approach to measure the coupling impedance using a Goubau line (G-line), which is essentially a single wire transmission line designed to propagate Sommerfeld-like surface waves whose fundamental Transverse Magnetic (TM) mode mimics the Coulomb field of a relativistic particle beam. We describe in detail the measurement procedure that we have developed for the G-line, including the measurement setup and proper definition of a reference, measurement procedure and advantages, and our experience regarding how to reduce systematic experimental error that we learned over the course of the measurements. Starting with our initial suite of measurements and simulations designed to benchmark and validate the novel G-line based measurement technique, we present the measured results for several Advanced Photon Source Upgrade vacuum components, including those of two rf-gasket designs and the beam position monitor-bellows assembly.