No, it is not possible to increase the dots per inch (DPI) of a cached map service for printing. This is by design. When creating a cached map service, the default tiling scheme is predefined and set to 96 DPI and cannot be changed.
However, below are several possible options for increasing the DPI for printing large, high quality maps.
Option 1: Enable dynamic layers in the cached map service
Dynamic layers enable users to change the properties of a layer in a cached map and allows the print service to obtain map images at a higher resolution. The following links describe how to enable dynamic layers in ArcGIS for Desktop and ArcGIS Server Manager.
Option 2: Generate a custom tiling scheme
The Generate Map Server Cache Tiling Scheme tool creates an XML file which defines the tiling scheme properties of a cached map service.
Check the following documentation for the list of map properties which can be changed: ArcGIS for Desktop: Generate Map Server Cache Tiling Scheme.
Follow the steps given to set the DPI in the map caching properties:
1. Open ArcMap.
2. Navigate to ArcToolbox > Server Tools > Caching, and double-click the Generate Map Server Cache Tiling Scheme tool.
3. Define the necessary settings, and set the desired DPI to be used.
4. Click OK to generate the tiling scheme file.
5. Navigate to File > Share As > Service, and republish the map document.
6. In the Service Editor window, navigate to Caching and for the Draw this map service option, select Using tiles from a cache. For the Tiling Scheme, click Suggest to reveal the drop-down menu, and select A tiling scheme file.
7. When prompted, navigate to the location of the saved XML file created in Step 4. Select the XML file, and click OK.
8. Click OK in the Service Editor to publish the service with the custom tiling scheme.
Once the service is published the DPI reflects the changes made in the XML file.
There are instances where custom tiling schemes do not align with other predefined tiling schemes because of dpi differences.
Option 3: Use the arcpy.mapping module
The arcpy.mapping Python module is designed to change the content of a map or layer and also provides functionality to automate exporting and printing. The following tutorial describes how to leverage the arcpy.mapping module: ArcGIS for Desktop: Tutorial: Basic web map printing and exporting using arcpy.mapping.