Mental deterioration or Alzheimer’s (ALZ) disease is progressive and causes both physical and mental dependency. There is a need for a computer-aided diagnosis (CAD) system that can help doctors make an immediate decision. (1) Background: Currently, CAD systems are developed based on hand-crafted features, machine learning (ML), and deep learning (DL) techniques. Those CAD systems frequently require domain-expert knowledge and massive datasets to extract deep features or model training, which causes problems with class imbalance and overfitting. Additionally, there are still manual approaches used by radiologists due to the lack of dataset availability and to train the model with cost-effective computation. Existing works rely on performance improvement by neglecting the problems of the limited dataset, high computational complexity, and unavailability of lightweight and efficient feature descriptors. (2) Methods: To address these issues, a new approach, CAD-ALZ, is developed by extracting deep features through a ConvMixer layer with a blockwise fine-tuning strategy on a very small original dataset. At first, we apply the data augmentation method to images to increase the size of datasets. In this study, a blockwise fine-tuning strategy is employed on the ConvMixer model to detect robust features. Afterwards, a random forest (RF) is used to classify ALZ disease stages. (3) Results: The proposed CAD-ALZ model obtained significant results by using six evaluation metrics such as the F1-score, Kappa, accuracy, precision, sensitivity, and specificity. The CAD-ALZ model performed with a sensitivity of 99.69% and an F1-score of 99.61%. (4) Conclusions: The suggested CAD-ALZ approach is a potential technique for clinical use and computational efficiency compared to state-of-the-art approaches. The CAD-ALZ model code is freely available on GitHub for the scientific community.