Radar systems for direction of arrival (DoA) estimation have been the subject of significant research with applications ranging from security to channel sounding and automotive radars. Conventional DoA retrieval techniques rely on an array based system architecture as the receiving unit, typically synthesized at the Nyquist limit. This classical array based approach makes it necessary to collect the received radar signals from multiple channels, and process it using DoA estimation algorithms to retrieve the DoA information of incoming far-field sources. A challenge with this multi-pixel approach is that, as the operating frequency is increased, the number of antennas (and hence the number of data acquisition channels) also increases. This can result in a rather complex system architecture at the receiver unit, especially at millimetre-wave and submillimetre-wave frequencies. As an enabling technology for the compressing sensing paradigm, a single-pixel based coded aperture can substantially simplify the physical hardware layer for DoA estimation. A significant advantage of this technique is that the received data from the source is compressed into a single channel, circumventing the necessity to have array-based multiple channels to retrieve the DoA information. In this work, we present a passive compressive sensing radar technique for DoA estimation using a single-frequency, dynamically reconfigurable wave-chaotic metasurface antenna as a receiver. We demonstrate that using spatiotemporarily incoherent measurement modes generated by the coded programmable metasurface aperture to encode and compress source generated far-field incident waves into a single channel, we can retrieve high fidelity DoA patterns from compressed measurements.