Recent advancements in CubeSat technology unfold new mission ideas and the opportunity to lower the cost of space exploration. Ground operations costs for interplanetary CubeSats, however, still represent a challenge towards low-cost CubeSat missions: hence, certain levels of autonomy are desirable. The feasibility of autonomous asteroid flyby missions using CubeSats is assessed here, and an effective strategy for autonomous operations is proposed. The navigation strategy is composed of observations of the Sun, visible planets, and the target asteroid, whereas the guidance strategy is composed of two optimally-timed trajectory correction maneuvers. A Monte Carlo analysis is performed to understand the flyby accuracies that can be achieved by autonomous CubeSats, in consideration of errors and uncertainties in: (a) departure conditions, (b) propulsive maneuvers, (c) observations, and (d) asteroid ephemerides. Flyby accuracies better than ±100 km (3σ) are found possible, and main limiting factors to autonomous missions are identified, namely: (a) on-board asteroid visibility time (Vlim≥11), (b) ΔV for correction maneuvers (>15 m/s), (c) asteroid ephemeris uncertainty (<1000 km), and (d) short duration of transfer to asteroid. Ultimately, this study assesses the readiness level of current CubeSat technology to autonomously flyby near-Earth asteroids, in consideration of realistic system specifications, errors and uncertainties.