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If you have the backup archive you can restore your site. You do not need access to your site. First copy your backup archive from the backup directory to your site's root directory (typically called public_html, htdocs or something similar). Then follow the instructions in this video: https://www.akeebabackup.com/videos/1212-akeeba-backup-core/1618-abtc04-restore-site-new-server.html

Please remember that Akeeba Backup archives are self-sufficient. They include a copy of your site's files, your site's database and the restoration script. All you need to do is extract the archive. Moreover, they are designed to be restored anywhere: the same server, a different location on the same server, or even a completely different server. It doesn't matter where you restore them, the restoration itself is always the same. I wrote that because the video's title is "restoring a site on a new server" and you said you want it restored on the same server. No worries, it's the same instructions when you don't have access to your site's backend exactly because the restoration script doesn't care where you are restoring to. It's designed to work anywhere.