It seems like there are two questions here:
1: How could a moonless planet capture a single asteroid into a closed orbit around the planet?
In a 2-body orbital system, a moonless planet couldn't capture an asteroid. The asteroid would leave the planet's gravity well with the same relative speed that it entered.
However, a sun/planet/asteroid system is a 3-body orbital system. If an asteroid is moving from perihelion to aphelion as it passes through a planet's gravity well, it may be slowing down due to the sun's gravitational pull. If the asteroid's orbital velocity with respect to the planet drops enough, the asteroid will be captured.
Jupiter captures comets and asteroids all the time without significant interactions with its moons. There is a pretty good paper by Prado and Broucke called The capture of comets by swing-by describing Jupiter encounters that end up in closed Jupiter orbits. They define a point $X_c$ and an angle $\theta$ for a Jupiter/Sun line crossing:
Comets passing on points near Jupiter with low angles and low relative velocities with Jupiter have a higher probability of orbital capture.
2: How can a highly elliptical orbit of an asteroid around a moonless planet become circularized?
Tidal Circularization could decrease the eccentricity of a captured asteroid.
Example: If tidal locked, the asteroid will rotate at (very close to) a uniform rate, but won't revolve around the planet at a uniform rate. At it's fastest orbital speed at periapsis, the planet-side bulge pulls ahead and the opposite-side bulge lags behind. The planet-side bulge has more gravitational force exerted on it, which decelerates the asteroid slightly, decreasing the next apoapsis. A similar, but opposite effect occurs at apoapsis, increasing periapsis. When periapsis and apoapsis meet, the orbit is circularized. This would happen to our Moon, in theory, if it wasn't for the Sun's gravitational influence.
More discussion of tidal circularization is here:
Is the moon's orbit circularizing? Why does tidal heating circularize orbits?