Speed-of-sound (SoS) has been shown as a potential biomarker for breast cancer imaging, successfully differentiating malignant tumors from benign ones. SoS images can be reconstructed from time-of-flight measurements from ultrasound images acquired using conventional handheld ultrasound transducers. Variational networks (VNs) have recently been shown to be a potential learning-based approach for optimizing inverse problems in image reconstruction. Despite earlier promising results, these methods, however, do not generalize well from simulated to acquired data, due to the domain shift. In this work, we present for the first time a VN solution for a pulse-echo SoS image reconstruction problem using diverging waves with conventional transducers and single-sided tissue access. This is made possible by incorporating simulations with varying complexity into training. We use loop unrolling of gradient descent with momentum, with an exponentially weighted loss of outputs at each unrolled iteration in order to regularize the training. We learn norms as activation functions regularized to have smooth forms for robustness to input distribution variations. We evaluate reconstruction quality on the ray-based and full-wave simulations as well as on the tissue-mimicking phantom data, in comparison with a classical iterative [limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS)] optimization of this image reconstruction problem. We show that the proposed regularization techniques combined with multisource domain training yield substantial improvements in the domain adaptation capabilities of VN, reducing the median root mean squared error (RMSE) by 54% on a wave-based simulation data set compared to the baseline VN. We also show that on data acquired from a tissue-mimicking breast phantom, the proposed VN provides improved reconstruction in 12 ms.