Knowledge Graph (KG) embeddings are essential for link prediction over KGs. Compared to triplets, hyper-relational facts consisting of a base triplet and an arbitrary number of key-value pairs, can better characterize real-world facts and have aroused various hyper-relational embedding techniques recently. Nevertheless, existing works seldom consider the ontology of KGs, which is beneficial to link prediction tasks. A few studies attempt to incorporate the ontology information, by either utilizing the ontology as constraints on entity representations or jointly learning from hyper-relational facts and the ontology. However, existing approaches mostly overlook the ontology hierarchy and suffer from the dominance issue of facts over ontology, resulting in suboptimal performance. Against this background, we propose a universal contrastive learning framework for hyper-relational KG embeddings (HyperCL), which is flexible to integrate different hyper-relational KG embedding methods and effectively boost their link prediction performance. HyperCL designs relation-aware Graph Attention Networks to capture the hierarchical ontology and a concept-aware contrastive loss to alleviate the dominance issue. We evaluate HyperCL on three real-world datasets in different link prediction tasks. Experimental results show that HyperCL consistently boosts the performance of state-of-the-art baselines with an average improvement of 3.1-7.4% across the three datasets.