Interesting question. I assume with "bordering Med" you mean the Cilician Armenian kingdom which indeed was bordering the Mediterranean Sea? Or maybe during Tigran the Great's reign? Regardless, I'll explain in a bullet point style:

• Historically Armenia or Armenian Highlands has been defined as a region bordering Anatolia in the west and spanning east towards the lowlands of Kur(a)/Arax(a). This is quite a broad region that transitions to the Caucasus in the north. So, technically Armenia has always been around the Caucasus. Now, it is debatable if it was part of Caucasus, as Armenian Highlands constitutes a separate geographical region.
• During the last century, Western portion of Armenia has been consistently and deliberately been renamed to Eastern Anatolia: this is a whole other discussion. Importantly, the Eastern portion of Armenian Highlands where the current Armenian republic lies is nowadays considered part of South Caucasus.
• Now for the Mediterranean: until the Middle Ages, Armenia bordered the Mediterranean only once, during the short span of the Armenian "empire" under Tigranes the Great in the 1st century BC. He conquered mainly the mountainous Cilicia and Syria, but after a couple of decades, these territories passed to the Roman republic.
• Cilicia had some Armenian population from late Antiquity/early Middle Ages, but starting from the 10th/11th centuries due to Eastern Roman Empire's policies and afterward Turkic migration and incursions into Armenia proper, many common and noble Armenians congregated in Cilicia, a territory in southeastern Anatolia. Soon Armenians constituted a relative majority and established first a principality and then a kingdom (which was generally called "Little Armenia" or "Armenian Kingdom").
• This state had a very close relationship with the Crusader states and contributed greatly to the Christian efforts in the Outremer.
• The kingdom finally fell in the 14th century and was thoroughly devastated by the Mamluks. Nevertheless, Armenians still constituted a majority in some regions of Cilicia, especially in the mountains until the 20th century. Notably, Adana had a large Armenian population and saw the 1909 Adana Armenian massacres. In the mountains of Cilicia, Armenian Marash Zeytun was renowned for its ferocity and fighting spirit.
• During the Armenian Genocide the region, like elsewhere in the Ottoman Empire, was stripped of its Armenian population.
• After WWI, Cilicia was to become a French mandate, and with French troops, many Armenian refugees and soldiers returned to Cilicia. In fact, one Armenian delegation was advocating for the inclusion of Cilicia in a newly created Armenian republic, but the idea never caught up with the European powers.
• After French defeats at the hand of Atatürk's Turkish forces, Armenians tried to declare an independent republic, but were pushed back and partially massacred by Turkish troops and were forced to retreat. So ended the Armenian presence there.
• Although Cilicia was outside of traditional Armenian homeland, it is a vital piece of Armenian history and provided Armenians a unique opportunity to interact with many different cultures, most notably Western European. Armenian culture flourished and many great intellectuals worked there, for a time the of the Armenian church had his seat in Cilicia. Cilician Armenia had a very strong trade network and an established fleet in the Mediterranean Sea. Marco Polo passed through Armenian Cilicia on his voyage to China. Italian merchants had their own quarters in coastal cities and Crusader militant orders (Templars, Hospitaliers and Teutonic order) were given castles in Armenian Cilicia.
• Nowadays, the remains of Armenian churches and castles are the only material evidence of Armenian presence there. Many of Cilician Armenians later found refuge in Syria and Lebanon. But Cilicia still leaves in Armenian imagination in the image of for example folk songs.

Edit: small correction