Automated vehicles need to detect and classify objects and traffic participants accurately. Reliable object classification using automotive radar sensors has proved to be challenging. We propose a method that combines classical radar signal processing and Deep Learning algorithms. The range-azimuth information on the radar reflection level is used to extract a sparse region of interest from the range-Doppler spectrum. This is used as input to a neural network (NN) that classifies different types of stationary and moving objects. We present a hybrid model (DeepHybrid) that receives both radar spectra and reflection attributes as inputs, e.g. radar cross-section. Experiments show that this improves the classification performance compared to models using only spectra. Moreover, a neural architecture search (NAS) algorithm is applied to find a resource-efficient and high-performing NN. NAS yields an almost one order of magnitude smaller NN than the manually-designed one while preserving the accuracy. The proposed method can be used for example to improve automatic emergency braking or collision avoidance systems.