One-bit radar, adopting one-bit analog-to-digital converters to perform signal sampling and quantization, is a promising technology for low-cost and resource-limited applications. In this paper, effects of one-bit quantization on the linear signal processing for the one-bit linear frequency modulated pulse radar are studied and a target reconstruction approach is proposed. Firstly, characteristics of 3-order harmonics of the one-bit signal are analyzed. It is indicated that when conventional linear signal processing methods are carried out, 3-order harmonics may not only raise the detection background but also result in false alarms. Secondly, a two-stage approach jointly performing the linear and nonlinear signal processing is proposed to suppress harmonics and recover true targets. Specifically, in the first stage, the linear signal processing is carried out to suppress unmatched harmonics and dimension reduction. In the second stage, based on a dimension-reduced signal observation model, generalized approximate message passing is utilized to suppress matched harmonics and recover true targets. The effectiveness of the proposed approach is demonstrated by numerical results.