From here: https://sharepoint-learners.blogspot.com/2016/09/sharepoint-2013-approve-list-item.html
Work Statement: Approve all the list items
$site = new-object Microsoft.SharePoint.SPSite("http://site")
$web = $site.OpenWeb()
$list = $web.Lists["Posts"]
$items = $list.Items
foreach ($item in $items)
{
$item["_ModerationStatus"] = 0
$item.Update()
}
Work Statement: You want to approve a single list item
$site = new-object Microsoft.SharePoint.SPSite("http://site")
$web = $site.OpenWeb()
$list = $web.Lists["Posts"]
$items = $list.GetItemById(10)
$item["_ModerationStatus"] = 0
$item.Update()
The following are more ModertionStatus:
| Value | Description |
| 0 |
The list item is approved. |
| 1 |
The list item has been denied approval. |
| 2 |
The list item is pending approval. |
| 3 |
The list item is in the draft or checked out state. |
| 4 |
The list item is scheduled for automatic approval at a future date. |