Rendezvous and proximity operations (RPO) for satellites have seen rising interest in recent years, as the subject has many applications in the realm of defense, the emerging in-space economy, and human space exploration. One of the most promising recent developments in the field is the second-order cone programming (SOCP) formulation of the problem, which efficiently produces an optimal solution for highly-constrained missions on any arbitrary orbit. The efficiency of the SOCP method allows for autonomous closed loop trajectory optimization, greatly expanding the tolerance to modeling uncertainties. The SOCP RPO formulation poses the mission as a specified final time problem and requires that this important parameter be precomputed. An efficient method to estimate the optimal time of flight in real time is developed using machine learning techniques. This allows the parameter to be updated with each trajectory replanning step and may significantly reduce propellant usage during the mission. A thorough investigation of machine learning methods is completed for a sample RPO mission. A dataset of 15,000 variations on the same sample mission is produced. Classical feedforward neural networks, convolutional neural networks, and multiple linear regression methods are trained on this dataset, and accuracy comparisons are made. A wide variety of practical issues for machine learning methods applied to this problem are discussed and solved; this includes optimization of network topology, feature subset selection, and the dataset size requirements. Numerical demonstrations of sample cases using the neural network estimated time of flight show improvements in fuel consumption and some of the potential downfalls of a machine learning approach.