Do you have a child theme set up? Once you do that, put a copy of the header.php file in the child theme and add the image code just below this line.
<h2 class="site-description"><?php bloginfo( 'description' ); ?></h2>
You'll have to fill in the relevant info for your site:
<a href="url to page you are linking to" <img src="file path to your image" alt="put alt text here" width="xx" height="xx" class="aligncenter buttonimg" /></a>
It should be right above this line of code:
The above adds a class to that image so you can style it with CSS if you want:
styles go here;
Having said all that, I'm wondering if what you really want is to add or move the menu to that location on the page?