How to run a program as an administrator at startup on Windows 10?

Create a Scheduled Task to trigger at log on.

In the Create Task dialog, select the following:

1. General (tab), Run with highest privileges
2. Triggers (tab), New (button), Begin the task, At log on

This is a little old, but I was having the same problems, and none of the above was working. What I did was create a VBScript that ran at startup that opened said program.

1. Right-click on the program, go to properties, then compatibility and check "Run as Administrator"

2. Create the VBScript using a text editor (I use Notepad++)

Script:

Set WshShell = CreateObject("WScript.Shell" ) 
WshShell.Run """C:\Program Files (x86)\File\Program.exe""", 0 'Must quote command if it has spaces; must escape quotes
Set WshShell = Nothing

Note: that C:\Program Files (x86)\File\Program.exe is the full path to the program with extension. Also, make sure to save it as a .vbs

3. Now place the VBScript in the startup folder:

   %AppData%\Microsoft\Windows\Start Menu\Programs\Startup

^{Or, alternatively access it by Win+R ⇒ shell:startup ⇒ Enter}

I used this instead of a batch file because I did not want that ugly command window showing up.

Well why don't you simply copy the shortcut of the program to the Startup folder and on its properties, enable the administrator permissions on the Compatibility tab?