According to Wictor Wilén,

The Client Object Model is fairly limited when it comes to working with Web Parts. Basic operations such as adding and removing Web Parts can be done as well as changing some default properties of the Web Part (such as Title). There's no access to custom Web Part properties.

There's no access to custom Web Part properties. There's no way to access the custom properties or methods on a Web Part. If you need to do this kind of customization with remote clients, you have to add your own remote API to SharePoint.

For more details please click Here.

To get the required details of the webpart I used a combination of both methods. First, I fetched the list of web part details using the following code:

ClientContext ctx = new
ClientContext("http://foo");

File home = ctx.Web.GetFileByServerRelativeUrl("/SitePages/Page.aspx");

var wpm = home.GetLimitedWebPartManager(PersonalizationScope.Shared);

var query = wpm.WebParts.Include(wp => wp.Id, wp => wp.WebPart));

var webPartDefenitions = ctx.LoadQuery(query);

ctx.ExecuteQuery();

Then called the GetWebPart2 method of Webpartpages web service to download the content:

var client = new
WebPartPagesWebService();

client.Url = siteRootAddress+"/_vti_bin/Webpartpages.asmx";

client.Credentials = credential;

// webPartId is a property of WebPart Defenition from the above code

var webPartXmlString = client.GetWebPart2(pageAddress,

webPartId,

Storage.Shared,

SPWebServiceBehavior.Version3);

var webPartNode = XElement.Parse(webPartXmlString);