We propose a new method of nonparametric bootstrap to quantify estimation uncertainties in functions of network degree distribution in large ultra sparse networks. Both network degree distribution and network order are assumed to be unknown. The key idea is based on adaptation of the ``blocking'' argument, developed for bootstrapping of time series and re-tiling of spatial data, to random networks. We first sample a set of multiple ego networks of varying orders that form a patch, or a network block analogue, and then resample the data within patches. To select an optimal patch size, we develop a new computationally efficient and data-driven cross-validation algorithm. In our simulation study, we show that the new fast patchwork bootstrap (FPB) outperforms competing approaches by providing sharper and better calibrated confidence intervals for functions of a network degree distribution, including the cases of networks in an ultra sparse regime. We illustrate the FPB in application to analysis of social networks and discuss its potential utility for nonparametric anomaly detection and privacy-preserving data mining.