Note: This answer is written in the form of a fictional, in-universe briefing document. I've done my best to keep the "facts" correct and sourced from the Ringworld novels, but this "document" is my own invention. The format of this answer is currently under discussion in meta.
Navigating the Ringworld
NOTE: THIS DOCUMENT IS CONFIDENTIAL, AND THE PROPERTY OF THE UNITED NATIONS AMALGAMATED REGIONAL MILITIA. DISTRIBUTION IS PROHIBITED AND PUNISHABLE BY LAW.
Purpose and context
This document assumes the reader is familiar with the Ringworld, a megastructure located outside of Known Space; but in brief:
The "Ringworld" is an artificial ring about one million miles wide and approximately the diameter of Earth's orbit (which makes it about 600 million miles in circumference), encircling a Sol-type star. It rotates, providing an artificial gravity equivalent to 99.2% of Earth's gravity by way of centrifugal force. Ringworld has a habitable flat inner surface equivalent in area to approximately three million Earth-sized planets.
(Source of summary text, including Ringworld parameters)
There are many possibilities for an ARM agent to navigate the Ringworld. After the ARM's interaction with the Ringworld during the Fringe War, it became clear that any future exploratory forces that need to prepare to navigate on the Ringworld surface (should its location become known again) would need guidance in this regard. This article, assembled by the United Nations ARM police force, is an extrapolation based on the Ringworld's known features.
While conventional spacecraft autopilots suffice for navigating in space and around planets, the Ringworld presents special challenges, including the presence of an extremely potent defense system. You will not be able to easily use your ship for travel on the Ringworld without risking being shot down by the meteor defense system. Small craft like flycycles and landing boats can travel on the Ringworld safely, as long as their altitude is no higher than that of the ring walls (approximately 1,600km).
(Landing on the Ringworld is possible. Approved landing procedures are discussed in detail in the ARM document "Ringworld landing procedures", and difficulties are laid out in the summary of the 2855 expedition to the Ringworld.)
Electronic satellite navigation
The Ringworld's shadow squares are large panels of an unknown material, whose main purpose is to shade parts of the Ringworld from the sun and create a day/night cycle. 921
They are connected by a monofilament thread that, unlike Sinclair Molecule Chain, retains its cutting ability even when not at tension. The shadow squares are part of an extremely sophisticated system that was designed to expand and contract as needed, not only providing night and day but also being part of a correctional and defensive system.
The squares are linked electronically to the Ringworld repair center, and the ring surface is plainly visible from their location hugging the Ringworld's star. It has been confirmed that observation and tracking equipment is present and functional.
Assuming that the so-called "Ringworld engineers" kept and maintained accurate maps of the surface, a GPS-like system would have been be a relatively simple matter for them to implement on top of existing detecting equipment. However, such a system would likely not be available to an ARM agent. Early probes of the Ringworld showed that the shadow square system is now in some disrepair, but its defensive capabilities are confirmed as operational. These defensive measures, X-ray lasers powered with energy from the star's photosphere, are also most effective in the plane of the Ringworld itself. They can and do strike against targets on the surface of the ring. (See document "Ringworld defense system capabilities and avoidance" for critical information about this deterrent.)
Even if any shadow square navigation system is still functioning, it is almost certainly a complex and sensitive system with sophisticated firewalls. (See document: "Ringworld Engineer: Probable identity and protective nature".) If an agent were able to get past the software defenses, the risk of using this potential resource is very high.
Recommendation: Avoid
Scrith-based electronic navigation
There is a network of superconducting cable embedded in the ring floor. It has been shown that signals can travel through scrith (the Ringworld floor material), so it may be possible to use this network as a sort of location system.
Even if there are no active signals in the network, a sort of magnetic echolocation may be useful using deep radar.
Recommendation: Promising
Stellar navigation
When one is on the Ringworld the night sky is never fully dark - the arc of the ring overhead drowns out the dimmer stars. But one can see stars at night, so navigating by the stars is quite possible.
The visible constellations change with each rotation of the ring, just over nine days. 302