Name, Department, #of Emails, Mailbox Size

lpmdvip Here you have. 
About the lastuseractiontime property, I haven't used that one before and not have documentation on its purpose. 

 <#
.SYNOPSIS
    Get MBXs Stats, User Attribute(s), Folder(s) Counts
.DESCRIPTION
    
.NOTES
    Author:     Erick A. Moreno
    Email:      emoreno@request-script.com
    Date:       12 Jan 2020
    PowerShell: v3
.References
#requested: https://techcommunity.microsoft.com/t5/windows-powershell/name-department-of-emails-mailbox-size/m-p/1167911
#>

$CredentialsFolder = "$home\Documents\"
$WorkingDir        = "$home\Documents"
$ScriptTime        = Get-Date -UFormat "%m%d%Y-%H%M"

#region Functions
Function _Get-Sessions
 {
  Param(
        [Parameter(Mandatory=$true)]$CredentialsFolder,
        [Parameter(Mandatory=$true)]$CredentialsFileName
       )
  
  #region Check for Credentials File
  $TargetCredPath = $CredentialsFolder #Credential XML File Location
  $TargetCredName = $TargetCredPath + $CredentialsFileName
  #Write-Host $TargetCredName
  If(Test-Path $TargetCredName)
   {
    $TargetCred = Import-Clixml $TargetCredName
   }
  Else
   {
    $Cred = Get-Credential 
    $Cred | Export-CliXml -Path $(Join-Path -path $TargetCredPath -ChildPath RemotePS.cred)
    $TargetCred = Import-Clixml $TargetCredName
   }
  #endregion
  #region connections
  $Session  = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $TargetCred -Authentication Basic -AllowRedirection 
  $NoOutput = Import-PSSession $Session -AllowClobber -Prefix O365 -CommandName "*Mailbox*","*User*"
  #endregion
  Return  $Session
 }
#endregion
#region Open PS Sessions
$TenantSession = _Get-Sessions -CredentialsFolder $CredentialsFolder -CredentialsFileName RemotePS.cred
#endregion

$MBXs = Get-O365Mailbox -ResultSize Unlimited 

$Counter = $MBXs.count
$Results = @()
$NoDepartment = @()
$Counter

Foreach($MBX in $MBXs)
 {
  #Start-Sleep -Milliseconds 300
  $PsSessionState = $(Get-PsSession $TenantSession.Id).State  
  If($($Counter/100) -match '^[0-9]+$' -or $PsSessionState -ne "Opened")
   {
    Remove-PSSession $TenantSession.Id -Confirm:$false
    Write-Host "Opening a new Session Each 100 iterations" -ForegroundColor Yellow
    $TenantSession = _Get-Sessions -CredentialsFolder $CredentialsFolder -CredentialsFileName RemotePS.cred
   }

  Write-Host "($Counter) Working On MBX: $($MBX.PrimarySmtpAddress)" -ForegroundColor Cyan
  $Stats       = $null
  $UserData    = $null
  $FolderItems = $null
  $UserData    = Get-O365User $MBX.PrimarySmtpAddress

  If(![string]::IsNullOrEmpty($($UserData.Department)))
   {
    $Stats       = Get-O365MailboxStatistics -Identity $MBX.PrimarySmtpAddress -Archive 
    $FolderItems = Get-O365MailboxFolderStatistics -Identity $MBX.PrimarySmtpAddress | Where {$_.Name -match “Inbox|Sent Items|Deleted Items|Junk E-Mail”} | Select Name,ItemsInFolder #Add Folders as needed
    
    $Properties  = [Ordered]@{
                              DisplayName        = $MBX.DisplayName
                              Department         = $UserData.Department #Add properties as needed
                              UserPrincipalName  = $MBX.UserPrincipalName
                              MailboxSizeMB      = [math]::Round($($Stats.TotalItemSize.Value.ToString().Split("(")[1].Split(" ")[0].Replace(",","")/1MB),4)
                              MailboxSizeGB      = [math]::Round($($Stats.TotalItemSize.Value.ToString().Split("(")[1].Split(" ")[0].Replace(",","")/1GB),4)
                              InboxItems         = $($FolderItems | where{$_.Name -eq "Inbox"}).ItemsInFolder
                              SentItems          = $($FolderItems | where{$_.Name -eq "Sent Items"}).ItemsInFolder
                              DeletedItems       = $($FolderItems | where{$_.Name -eq "Deleted Items"}).ItemsInFolder
                              JunkItems          = $($FolderItems | where{$_.Name -eq "Junk E-Mail"}).ItemsInFolder
                              LastLogon          = $Stats.LastLogonTime.ToString()
                              lastuseractiontime = $Stats.lastuseractiontime 
                              PrimarySMTPAddress = $MBX.PrimarySmtpAddress
                             }
    
    $Results += New-Object PSObject -Property $Properties
   }
  Else
   {
    Write-Host "Department Field is Empty Skipping MBX: $($MBX.PrimarySmtpAddress)"
    $NoDepartment += $($MBX.PrimarySmtpAddress)
   }
  $Counter--
 }

 $Results | Export-Csv -Path "$WorkingDir\ReportMBXsStats_$ScriptTime.csv" -NoTypeInformation -Encoding UTF8 -Force
 $NoDepartment | Export-Csv -Path "$WorkingDir\NoDepartmentField_$ScriptTime.csv" -NoTypeInformation -Encoding UTF8 -Force
 Write-Host "Report Saved To: $WorkingDir\ReportMBXsStats_$ScriptTime.csv" -ForegroundColor Cyan
 Write-Host "Report Saved To: $WorkingDir\NoDepartmentField_$ScriptTime.csv" -ForegroundColor Cyan