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Age-of-Information (AoI) is a critical metric for network applications. Existing works mostly address optimization with homogeneous AoI requirements, which is different from practice. In this work, we optimize uplink scheduling for an access point (AP) over multiple unknown channels with heterogeneous AoI requirements defined by AoI-dependent costs. The AP serves $N$ users by using $M$ channels without the channel state information. Each channel serves only one user in each decision epoch. The optimization objective is to minimize the time-averaged AoI-dependent costs plus additional communication transmission costs over an infinite horizon. This decision-making problem can be formulated as a Markov decision process, but it is computationally intractable because the size of the state space grows exponentially with respect to the number of users. To alleviate the challenge, we reformulate the problem as a variant of the restless multi-armed bandit (RMAB) problem and leverage Whittle's index theory to design an index-based scheduling policy algorithm. We derive an analytic formula for the indices, which reduces the computational overhead and facilitates online adaptation. Our numerical examples show that our index-based scheduling policy achieves comparable performance to the optimal policy and outperforms several other heuristics.