Graph-structured data is ubiquitous in the world which models complex relationships between objects, enabling various Web applications. Daily influxes of unlabeled graph data on the Web offer immense potential for these applications. Graph self-supervised algorithms have achieved significant success in acquiring generic knowledge from abundant unlabeled graph data. These pre-trained models can be applied to various downstream Web applications, saving training time and improving downstream (target) performance. However, different graphs, even across seemingly similar domains, can differ significantly in terms of attribute semantics, posing difficulties, if not infeasibility, for transferring the pre-trained models to downstream tasks. Concretely speaking, for example, the additional task-specific node information in downstream tasks (specificity) is usually deliberately omitted so that the pre-trained representation (transferability) can be leveraged. The trade-off as such is termed as "transferability-specificity dilemma" in this work. To address this challenge, we introduce an innovative deployment module coined as GraphControl, motivated by ControlNet, to realize better graph domain transfer learning. Specifically, by leveraging universal structural pre-trained models and GraphControl, we align the input space across various graphs and incorporate unique characteristics of target data as conditional inputs. These conditions will be progressively integrated into the model during fine-tuning or prompt tuning through ControlNet, facilitating personalized deployment. Extensive experiments show that our method significantly enhances the adaptability of pre-trained models on target attributed datasets, achieving 1.4-3x performance gain. Furthermore, it outperforms training-from-scratch methods on target data with a comparable margin and exhibits faster convergence.