Even before large expeditions into Brazil began, granting access to the New World's natural resources, the Portuguese were sending ships around both West and East Africa as well as India, Indonesia, and Japan. How was this possible while directly controlling such a small amount of land compared to Spain, Britain, or even the Ottomans? Did this naval focus strain the Portuguese economy more than the other European great powers?
First of all you need to understand that Portugal as always had a naval focus, since its foundation. The Portuguese navy has existed more or less continuosly for the past 900 years, it first engagement was in 1128 in Minho river. It was the first organized and professionalized section of the Portuguese armed forces, and has received the most focus throughout almost the entirety of Portuguese history.
The focus on navy only increased with the ending of the reconquista in Portuguese territory, and the begining to the age of discovery in 1415 with the conquest of ceuta. The wars with Castile in order to secure the Guinea trade, the thousands of trips to africa and elsewhere, the superior training due to the school in Sagres, as well as the enormous technological advancements created, introduced and put into widespread use by the sucessive kings and rulers made the Portuguese navy important and unmatched.
It's not unfair to say that by the time the Portuguese fleet of Vasco da gama arrived in India, Portugal was the foremost naval power in the world, if not in terms of numbers, in terms of technology and firepower.
The focus on naval power was also very much related to the lower population compared to places like castile or france. Ships are incredible force multipliers, and have uses beyond pure warfare, unlike land armies. This meant that expenses used on military ships, could be recouped by using them to carry spices or conduct trade.
This does not mean that ships were without cost. This book for example lists a number of subsidies given by the crown in order to encourage and aid ship construction and upkeep And there were a number of projects (like creating forests to sustain ship building) that were done throughout the centuries in order to assist shipbuilding.
Indeed shipbuilding has remained an important part of the Portuguese economy up until very recently (only after the entry into the european union have we lost that capacity).
TLDR: Long standing navy with lots of royal support, new technologies and experience was used to make up other shortcomings.
Adding to the reply by poloport, the development of naval power and (some) naval supremacy didn’t strain the economy but instead it was the driving engine for the economy.
In the early 13th century, Portuguese merchants were present in many European ports alongside Venetians, Genoese and others. Even the Portuguese crown, at times, engaged actively in equipping ships and taking part in sea trade. Sea trade to both the North Sea and the Mediterranean had already been important during roman times and significant during the middle ages, under Islamic rule, prior to the Reconquista. Hence, from pretty much the start of the country, seafaring was (still is) a significant facet in the national identity. As an illustrative point, Firshermen and Knights are depicted at roughly the same degree of importance in these very famous panels.
Furthermore, Portuguese harbors typically face onshore winds and outgoing ships are often forced to sail in open ocean conditions right from the start, and the major exports from Portugal were bulk goods like salt and agricultural products, which required large hulls. These geographic conditions, alongside the trading tradition, created the perfect mix for a sort of focused R&D effort, which the “School of Sagres” and Henry the Navigator embody.
In the XVth century Portuguese craftsmen were building carracks (“caravelas”), which were reputedly the fastest and most maneuverable ships in Europe. It was strictly forbidden to sell these ships to foreigners, and there are both a few examples of exceptions in royal edicts, usually as part of diplomatic negotiations, and of court cases where offenders were convicted and made penniless.
Portuguese kings placed themselves in the forefront of a “tech race”. They had massive profits from trade with the western African coast and wished to maintain monopoly. One of the strategies to achieve that monopoly was through Papal sanctioning (and it helped a lot that muslim interests got hurt by doing business directly at Africa’s coast). The other strategy was to be the only one able to sail there or to reach that far. This is why the investment was so significant, but at the same time, it was the reward from that expenditure that paid for further investment.