En route travel time estimation (ER-TTE) aims to predict the travel time on the remaining route. Since the traveled and remaining parts of a trip usually have some common characteristics like driving speed, it is desirable to explore these characteristics for improved performance via effective adaptation. This yet faces the severe problem of data sparsity due to the few sampled points in a traveled partial trajectory. Since trajectories with different contextual information tend to have different characteristics, the existing meta-learning method for ER-TTE cannot fit each trajectory well because it uses the same model for all trajectories. To this end, we propose a novel adaptive meta-learning model called MetaER-TTE. Particularly, we utilize soft-clustering and derive cluster-aware initialized parameters to better transfer the shared knowledge across trajectories with similar contextual information. In addition, we adopt a distribution-aware approach for adaptive learning rate optimization, so as to avoid task-overfitting which will occur when guiding the initial parameters with a fixed learning rate for tasks under imbalanced distribution. Finally, we conduct comprehensive experiments to demonstrate the superiority of MetaER-TTE.