How did the Axis intend to hold the Caucasus?

Not really important to hold. All the Soviet refineries were there. So it would slow down the war on both sides if those were lost.

Oil was only a minor problem, the real issue was climate. Germans were suffering various diseases;

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So there were many reasons, logistics, diseases climate, and it wasn't in any way useful for Germany to occupy the territory. That's why there was no plan. At best it could destroy the infrastructure and slow down the Soviets.

Once the Battle of Britain was lost and Germany failed to invade Britain the war was destined to be lost. The preconditions for Germany to win WW2 were Destroy the Royal Navy Take Gibraltar Take Malta Take North Africa Take the Middle East including Iraq and Iran Launch simultaneous attacks on USSR from the west and south and have Japan attack USSR from the east and avoid conflict with USA

The Germans were hoping that the Caucasus offensive would cripple the Red Army.

The Baku, Grozny, and Maikop oilfields together produced 86 percent of Russia's oil; Baku by itself, 72 percent. So the German plan was to capture these oilfields.

They captured Maikop and came close to capturing Grozny, but the path from Grozny to Baku was some 300 miles of mountains. (Further north, the Germans made progress by weaving "in and out" of mountains.)

The Germans might have done better to interdict the flow of Russian oil, by having Army group A advance to Astrakhan on the Caspian Sea, (and the lower Volga river between that city and Stalingrad.) That would have cut off the Caucasus and the Caspian Sea from the rest of Russia as far west as the Volga, and seriously impeded any flow even on the eastern side of the Caspian Sea. Perhaps the Russians would have to re-route their Baku oil through Iran and Turkestan, with little water transport along the way.

In any event, the two German Army Groups, B at Stalingrad and A at Astrakhan would have been mutually supporting.

Capturing and holding the resources of the Caucasus might have been too much to ask (initially). By denying those resources to the Russians, the Germans would probably have weakened them more than they would have been weakened themselves. Then it might have been possible to capture and hold those resources from a weakened Russia.

This is detailed in Führer directive 45.
It was incredibly easy. By just winning the war.

Details for this were variously named Fall Blau, Operation Brunswick, Operation Edelweiss…

As you see, there were not very many directives, and those pertaining to this plan were – ehm – optimistic.

Basically they "hoped to achieve this" with 'go in, smash the enemy on road to Caucasus, protect the flank by smashing the Reds at Stalingrad, then go to Iraq, Iran, India…'

No problem, just do it. The German soldier does as he is ordered, and the order was 'advance and win'. That would then all be 'automatic', as the Germans thought reaching oil fields meant immediate & unlimited supply of kerosene for their war effort and no more resources for Stalin after a few Kesselschlachten exterminating 'some' Soviet armies Cannae-style.

Hitler promised that both the Romanian mountain corps and three Italian mountain divisions would be committed to reinforce Heeresgruppe A by mid-August, enabling an equally rapid thrust to seize the Caucasian mountain passes and begin clearing the Black Sea coast. Edelweiss also made extensive provision for the use of German special forces to seize or sabotage key targets and Hitler authorised Heeresgruppe A to consider using airborne troops if feasible. However, the plan did not detail how German forces would reach distant Baku or what the Luftwaffe was expected to accomplish beyond supporting the army and attacking coastal shipping. At best, Edelweiss was an unfinished sketch, vulnerable to diverging objectives, limited knowledge of the terrain and the Führer’s whimsy.
_{–– Robert Forczyk: "The Caucasus 1942–43. Kleist’s race for oil", Osprey: London, 2015.}

However difficult to believe this may sound, but 'the grand design' really wasn't 'a grand design' – more a rough outline, for lack of detail. What will be will be.

The plan for a major offensive into the Caucasus to seize the oilfields was, to a much greater extent than the previous year's attack on the Soviet capital, Hitler's own strategic conception. Keitel, who thought the plan had considerable merit, wrote in his memoirs that the Führer 'conceived the idea entirely alone'. During the height of the winter crisis Hitler had unfairly but repeatedly cursed the General Staff for having imposed its Moscow campaign on him. Now that he had pulled Germany back from the brink of disaster he was determined to trust his instincts and order a campaign to attain his own strategic objectives (which were clearly shaped by his awareness of the Reich's economic problems). Moreover, he would no longer limit himself to issuing general instructions, but would, in his new capacity as Commander-in-Chief of the Army (since von Brauchitsch's resignation on 19 December), take complete and immediate charge of the direction of operations."
_{–– Joel Hayward: "Hitler's Quest for Oil: the Impact of Economic Considerations on Military Strategy, 1941–42", Journal of Strategic Studies, Vol.18, No.4 (December 1995), pp.94–135. (PDF)}

It is difficult to examine German military strategy as an influence on the direction taken by German covert operations in the Persian theatre and on the Allied response to them, because there was essentially no grand strategic concept in Berlin, from the very beginning to the bitter end of the Second World War. Indeed, there was in Nazi Germany no single, central military authority that could have worked out and coordinated an overall strategy. What passed for strategy was in fact a haphazard series of empirical judgements and situational responses, either steeped in political ideology or based on nothing more than operational pragmatism on the part of Adolf Hitler, a self-appointed, dilettantist military commander, not a trained strategist. Therefore, in this context, the term strategy describes nothing grander than the planning of operations at the army, corps, and divisional levels, and might be seen by some as synonymous with the term operational strategy or even operational tactics.
_{–– Adrian O’Sullivan: 'Schemers and Planners' in: "Nazi Secret Warfare in Occupied Persia (Iran). The Failure of the German Intelligence Services, 1939–45", Palgrave Macmillan: Basingstoke, New York, 2014.}

This imagined decisiveness is illustrated in how Hitler himself presented the lack of need for 'holding the Caucasus':

Had Hitler - as mentioned - still stated on April 30, 1942 that the war could "suddenly be over" with a successful advance over the Caucasus to Iran, an expectation which he had again expressed on April 4, 1942 during the initially successful German advance over the Don in the direction of the Caucasus. The events of the next six weeks on the Russian theater of war would be decisive for the war at all, so the leadership crisis triggered by the failure of the Caucasus operation at the beginning of September had marked the turnaround that had become generally recognizable externally in the transition of the initiative to the opponents in the East. Hitler's "fanatical" determination, without even considering the possibility of a separate peace, as sounded out by Stalin, to continue the racist ideology of the extermination war in the East in undiminished severity, was reflected in his remarks on Mussert, which were characterized by primitivity and brutality. Mussert's concern to gain clarity about Germany's intentions towards the Netherlands was only mentioned in passing and in a relatively vague form. As Hitler stressed, the "Great Germanic Empire" was to be created as a "secure, firmly established construction against the future Eastern storms".
_{–– Andreas Hillgruber & Jürgen Förster: "Zwei neue Aufzeichnungen über "Führer"-Besprechungen aus dem Jahre 1942", Militärgeschichtliche Mitteilungen; Freiburg Bd. 0, Ausg. 1, (Jan 1, 1972): 109.}

Or, as Manstein claims to remember and judges:

After removing Field-Marshal List from this appointment without valid reason, following a difference of opinion with him, Hitler had been commanding the Army Group himself as a sort of sideline - a quite impossible arrangement in the long run. More surprising still was what he had to say on this occasion in connexion with my eventual appointment as Commander-in-Chief of the Army Group. Next year, he told me, he was thinking of driving through the Caucasus to the Near East with a motorized army group! It was a measure of how unrealistically he still assessed the overall military situation and its strategic possibilities.

While Hitler may have had an eye for tactical opportunity and could quickly seize a chance when it was offered to him, he still lacked the ability to assess the prerequisites and practicability of a plan of operations. He failed to understand that the objectives and ultimate scope of an operation must be in direct proportion to the time and forces needed to carry it out — to say nothing of the possibilities of supply. He did not - or would not - realize that any long-range offensive operation calls for a steady build-up of troops over and above those committed in the original assault. All this was brought out with striking clarity in the planning and execution of the 1942 summer offensive. Another example was the fantastic idea he disclosed to me in autumn 1942 of driving through the Caucasus to the Near East and India with a motorized army group.

As in the political sphere (at all events after his successes of 1938), so in the military did Hitler lack all sense of judgement regarding what could be achieved and what could not. In autumn 1939, despite his contempt for France's powers of resistance, he had not originally recognized the possibility of attaining decisive success by a correctly planned German offensive. Yet when this success actually became his, he lost his eye for opportunity where conditions were different. What he lacked in each case was a real training in strategy and grand tactics.
_{–– Erich v Manstein: "Lost Victories, The War Memoirs of Hitler's Most Brilliant General" (Zenith Military Classics), 2004.}

More details in
_{–– U.S. Department of Defense: "German Campaign in Russia: Planning and Operations (1940-1942): WW2: Strategic & Operational Planning: Directive Barbarossa, The Initial Operations, German Attack on Moscow, Offensive in the Caucasus & Battle for Stalingrad", 2018.}