Machine learning under limited computational resources has gained increasing attention recently. A common yet challenging scenario is managing multiple time-constrained learning tasks with budgeted computational resources, known as Computational Resource Efficient Learning (CoRE-Learning). To this end, a recently proposed framework, Learning with Adaptive Resource Allocation (LARA), offers a preliminary approach. In this paper, we point out the limitations of LARA, including its reliance on interpolation-based extrapolation methods, the need for a fixed exploration phase, and the use of high-frequency re-estimation and reallocation strategies. To address these issues, we propose Look-ahead and immediate Resource Allocation (LaiRA). Our approach incorporates an efficient Dynamic Kalman Filtering (DKF) for look-ahead feasibility check with limited data and a weight-based online estimator for immediate performance evaluation. For resource allocation, LaiRA constructs an Upper Confidence Bound (UCB) to enable adaptive exploration and introduces an adaptive time-slicing method to reduce task switching costs. Empirical studies validate the effectiveness of our approach.