How to open Command Prompt in a specific folder as Administrator?

If you want the reasoning behind it, the Start in is explicitly ignored when elevation is performed (only on binaries that are part of Windows itself) to protect against a potential security vulnerability.

The basic idea is that potentially-malicious DLLs located in the working directory might be controlled by a user other than the current admin, and can then be loaded with high privileges. To prevent this, UAC will reset the working directory. Because shortcuts' "Start in" is set before elevation occurs, this gets reset during elevation. In an ideal world, this protection would apply to all elevations, but it only applies to built-in Windows binaries because it breaks some third-party programs that expect the working directory to be preserved.

The other answers bypass this by telling the elevated cmd to change its working directory after elevation occurs, via the /k argument.

In the target you should specify cmd /k cd c:\crp

Inside Explorer, there is a Quick Access Toolbar (QAT) since the intruduction of the Ribbon in Windows 8. Click on File->open CMD prompt->Open CMD prompt as admin and make a rightclick and select to pin it to QAT.

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Now you can click on this icon in every folder you like and the cmd now opens in this folder.

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To run the cmd as admin faster from the QAT, press the ALT key and you see a number for the position in the QAT.

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If you now press the number the tool at this position is started (in my case 4 runs the cmd as admin).