If you're using a layout software you would only need to be concerned about the DPI of any raster objects you use in the layout, with text and lineart being exempt from DPI issues due to their vector nature. When exporting your document as a PDF or sending it to a desktop printer, any raster objects whose effective output DPI is higher than the quality settings will be automatically resized to meet those requirements (in most cases).
However, it sounds like your designer built the file in Photoshop or GIMP - which poses a different issue entirely. While raster software is fine for making print quality photos, they aren't great for creating high-quality text without some finagling.
It's hard to give you a definite answer without knowing the actual dimensions of the label and an example of the artwork in question, so I'm going to base my answer off of some generalizations.
So is a 300DPI document good enough for a high quality print? Maybe, but probably not since it's a small label. It really depends on the nature and usage context of the artwork. In this context, tiny label = tiny text and fine detail, so 300 probably won't be enough. For reference, the software used to make printing plates for a professional press will typically rasterize outline shapes and text objects to at least 1000DPI.
If you want to avoid asking the designer to rework the document, I would recommend
submitting a NON-FLATTENED working document to your printer and asking if they think that it will print okay before proceeding. Remember to include any fonts used with the document. If I were in your position I would just skip this and ask the designer to rebuild it in proper layout software like InDesign.
(EDIT: There's further issues about rasterizing outlines and text with regards to the Trapping process in a commercial print environment that I've addressed in the comments. Also - since I last had to deal with this issue, Adobe changed the Photoshop PDF export to not rasterize text layers in the resulting document. This mitigates pretty much all of my concerns about type quality, so theoretically you can get away with using Photoshop to develop print documents... but I doubt you're going to find a printer who would be happy about it. Just don't do it.)