Is it possible to run clustering methods by only knowing the distance between pair of points?
K-Medoids

It would be possible with an adapted semi-supervised K-Means, also known as K-Medoids.

The tricky part with K-Means is that you do not know the centroids. However, you could hot start by assuming that some of your data points are centroids. Then, when figuring the new centroid at each iteration, instead of figuring out the "imaginary" central position, pick the point in the cluster that minimizes the sum to all other points in the cluster.

Hierarchical Clustering

You could also try a hierarchical clustering method. One example is the AgglomerativeClustering from Scikit-Learn. The idea is that you start merging points that minimize their linkage distance. Then, there is a certain criteria for determining points that are "too far away" and a new cluster should be created.

This is the fit() method documentation. Notice how you can either pass the instance features or the distance matrix between instances.