Deep Learning (DL) methods can reconstruct highly accelerated magnetic resonance imaging (MRI) scans, but they rely on application-specific large training datasets and often generalize poorly to out-of-distribution data. Self-supervised deep learning algorithms perform scan-specific reconstructions, but still require complicated hyperparameter tuning based on the acquisition and often offer limited acceleration. This work develops a bilevel-optimized implicit neural representation (INR) approach for scan-specific MRI reconstruction. The method automatically optimizes the hyperparameters for a given acquisition protocol, enabling a tailored reconstruction without training data. The proposed algorithm uses Gaussian process regression to optimize INR hyperparameters, accommodating various acquisitions. The INR includes a trainable positional encoder for high-dimensional feature embedding and a small multilayer perceptron for decoding. The bilevel optimization is computationally efficient, requiring only a few minutes per typical 2D Cartesian scan. On scanner hardware, the subsequent scan-specific reconstruction-using offline-optimized hyperparameters-is completed within seconds and achieves improved image quality compared to previous model-based and self-supervised learning methods.