Is there a way to write a rest controller to upload file using spring-data-rest without using Spring-MVC?

Spring Data Rest simply exposes your Spring Data repositories as REST services. The supported media types are application/hal+json and application/json.

The customizations you can do to Spring Data Rest are listed here: Customizing Spring Data REST.

If you want to perform any other operation you need to write a separate controller (following example from Uploading Files):

import java.io.BufferedOutputStream;
import java.io.File;
import java.io.FileOutputStream;

import org.springframework.stereotype.Controller;
import org.springframework.web.bind.annotation.RequestMapping;
import org.springframework.web.bind.annotation.RequestMethod;
import org.springframework.web.bind.annotation.RequestParam;
import org.springframework.web.bind.annotation.ResponseBody;
import org.springframework.web.multipart.MultipartFile;

@Controller
public class FileUploadController {

    @RequestMapping(value="/upload", method=RequestMethod.GET)
    public @ResponseBody String provideUploadInfo() {
        return "You can upload a file by posting to this same URL.";
    }

    @RequestMapping(value="/upload", method=RequestMethod.POST)
    public @ResponseBody String handleFileUpload(@RequestParam("name") String name,
            @RequestParam("file") MultipartFile file){
        if (!file.isEmpty()) {
            try {
                byte[] bytes = file.getBytes();
                BufferedOutputStream stream =
                        new BufferedOutputStream(new FileOutputStream(new File(name)));
                stream.write(bytes);
                stream.close();
                return "You successfully uploaded " + name + "!";
            } catch (Exception e) {
                return "You failed to upload " + name + " => " + e.getMessage();
            }
        } else {
            return "You failed to upload " + name + " because the file was empty.";
        }
    }

}

Yes you can try this:

@RestController
@EnableAutoConfiguration
@RequestMapping(value = "/file-management")
@Api(value = "/file-management", description = "Services for file management.")
public class FileUploadController {
    private static final Logger LOGGER = LoggerFactory
            .getLogger(FileUploadController.class);
    @Autowired
    private StorageService storageService;  //custom class to handle upload.
    @RequestMapping(method = RequestMethod.POST, headers = ("content-    type=multipart/*"), produces = "application/json", consumes =            MediaType.APPLICATION_FORM_URLENCODED_VALUE)
    @ResponseBody
    @ResponseStatus(value = HttpStatus.CREATED)
    public void handleFileUpload(
            @RequestPart(required = true) MultipartFile file) {
        storageService.store(file);  //your service to hadle upload.
    }
}