You can try making an Image object. Then in the src put the base64.

var myImage = new Image();
myImage.src="data:image/png;base64,"+file; 

In Imagemagick, you can do that with the INLINE: input structure. See http://www.imagemagick.org/Usage/files/#inline. For example:

convert 'inline:data:image/png;base64,
R0lGODlhIAAgAPIEAAAAAB6Q/76+vvXes////wAAAAAAAAAAACH5BAEAAAUALAAA
AAAgACAAAAOBWLrc/jDKCYG1NBcwegeaxHkeGD4j+Z1OWl4Yu6mAYAu1ebpwL/OE
YCDA0YWAQuJqRwsSeEyaRTUwTlxUqjUymmZpmeI3u62Mv+XWmUzBrpeit7YtB1/r
pTAefv942UcXVX9+MjNVfheGCl18i4ddjwwpPjEslFKDUWeRGj2fnw0JADs=
'  b64_noseguy.png

So this is your current code:

byte[] PDFDecoded = Convert.FromBase64String(base64BinaryStr);
string FileName = (@"C:\Users\Downloads\PDF " + DateTime.Now.ToString("dd-MM-yyyy-hh-mm"));

BinaryWriter writer = new BinaryWriter(File.Create(FileName + ".pdf"));            
writer.Write(PDFDecoded);

You don't actually need BinaryWriter for this. File.Create already gives you a FileStream:

FileStream writer = File.Create(FileName + ".pdf");
writer.Write(PDFDecoded, 0, PDFDecoded.Length);

But this will still have the problem you're experiencing because you're not flushing the data to it. We also need to close the file. Thankfully, we can wrap it in using and it will do both for us:

using (FileStream writer = File.Create(FileName + ".pdf"))
{
    writer.Write(PDFDecoded, 0, PDFDecoded.Length); 
}

But a simpler way to do this is:

File.WriteAllBytes(FileName + ".pdf", PDFDecoded);

As for PDF -> Image, you'll probably have to see if there is a library available for this (search "PDF to Image NuGet") that can help you with this as I don't think there is anything built-in.

While the comments suggested a TCPDF solution, I did want to state that there is a way to do this entirely with FPDF. The thought process is like this:

1. Strip data:image/png;base64, from the URI using explode
2. Decode the URI with base64_decode
3. Save the decoded data to a PNG file with file_put_contents
4. Generate a PDF and add the PNG file with Image
5. Delete the PNG file with unlink

Always remember to error check. If the image fails to save to the server you obviously do not want to continue execution. Make sure to always strictly check if the functions return false!

Here is my solution. I used a tiny image for a small URI in this example.

const TEMPIMGLOC = 'tempimg.png';

$dataURI    = "data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAPCAMAAADarb8dAAAABlBMVEUAAADtHCTeKUOwAAAAF0lEQVR4AWOgAWBE4zISkMbDZQRyaQkABl4ADHmgWUYAAAAASUVORK5CYII=";
$dataPieces = explode(',',$dataURI);
$encodedImg = $dataPieces[1];
$decodedImg = base64_decode($encodedImg);

//  Check if image was properly decoded
if( $decodedImg!==false )
{
    //  Save image to a temporary location
    if( file_put_contents(TEMPIMGLOC,$decodedImg)!==false )
    {
        //  Open new PDF document and print image
        $pdf = new FPDF();
        $pdf->AddPage();
        $pdf->Image(TEMPIMGLOC);
        $pdf->Output();

        //  Delete image from server
        unlink(TEMPIMGLOC);
    }
}