Here's a chart showing how the Ring will accelerate due to the Sun's gravity. As the Ring gets further and further off-center, the acceleration increases more and more. At first, the acceleration is pretty close to proportional to the displacement (acceleration in meters/sec/sec equals 2 times 10^(-14)/s/s times the displacement in meters), which means that it's easy to solve for the motion of the Ring - whatever the offset is, it will double every 56.5 days (until the offset gets to about 30% of the Ring diameter, after which things really start to go fast). So if the initial offset is one inch, in five years, the offset is 40,000 miles, and one year later, it's more than 3.5 million miles, and not that many months later, the warranty on your Ringworld is void (read the fine print!). It's even worse if there's an initial velocity too. In Ringworld Engineers, two characters estimate the time to collide with the Sun as being 5 years from the time of the initial disturbance - and that's a very reasonable estimate according to my calculations.