SharePoint 2010 CSOM library didn't have the property "$webpart.WebPart.Proerties", if you can access the SharePoint Server, use SharePoint Server Object Model in PowerShell instead:
Add-PSSnapin Microsoft.SharePoint.PowerShell
$SiteUrl = "http://www.yoursite.com"
$pageURL = "Pages/default.aspx"
$site = New-Object Microsoft.SharePoint.SPSite($SiteUrl)
$psite = New-Object Microsoft.SharePoint.Publishing.PublishingSite($site)
$web = Get-SPWeb $SiteUrl
$pweb = [Microsoft.SharePoint.Publishing.PublishingWeb]::GetPublishingWeb($web)
$wpm = $web.GetLimitedWebPartManager($pageURL, [System.Web.UI.WebControls.WebParts.PersonalizationScope]::Shared)
#List all the web parts
$wpm.WebParts | ft
#Get the details of the first web part
$wp = $wpm.WebParts[0]
$wp
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