The short answer is yes, the British monarch was technically the Head of State for South West Africa (formerly Deutsch-Südwestafrika) from the grant of the Mandate by the League of Nations until South Africa became a republic on 31 May 1961.
It's worth noting that South West Africa didn't become known as Namibia until the UN General Assembly changed the territory's name by Resolution 2372 (XXII) on 12 June 1968.
Article 22 of The Covenant of the League of Nations stated that:
There are territories, such as South-West Africa and certain of the South Pacific Islands, which, owing to the sparseness of their population, or their small size, or their remoteness from the centres of civilisation, or their geographical contiguity to the territory of the Mandatory, and other circumstances, can be best administered under the laws of the Mandatory as integral portions of its territory, subject to the safeguards above mentioned in the interests of the indigenous population.
(my emphasis)
The mandate for South-West Africa was granted to the Union of South Africa. Note that although not annexed to South Africa, South West Africa was "administered under the laws of" South Africa "as integral portions of its territory". Under Article 22, the de jure status of SW Africa was made explicitly part of South Africa for the duration of the mandate.
Unlike other Class-C mandate countries, South-West Africa did not become a United Nations trust territory in 1946 after South Africa objected. That objection was upheld by the International Court of Justice in 1950. However, a request by the South African government that the territory of South-West Africa should be annexed by South Africa as a fifth province was formally declined. See The South West Africa/Namibia Dispute: Documents and Scholarly Writings on the Controversy Between South Africa and the United Nations by John Dugard (University of California Press, 1973) p124 for more details
As a result, South-West Africa continued to be administered under the laws of South Africa as integral portions of its territory as established by Article 22 above. The mandate was not terminated until a UN General Assembly vote in 1966 (see below).
All this meant that the Head of State of South Africa was also technically the Head of State of South-West Africa, from the grant of the mandate in 1919 until it was formally repealed by the UN General Assembly in 1966 (resolution 2145 (XXI)).
The British monarch was the Head of State in South Africa until it became a republic, with the passage of the Republic of South Africa Constitution Act, 1961. Queen Elizabeth II became the monarch on 6 February 1952 and remained Head of State in South Africa until 31 May 1961 when South Africa became a republic, so the answer to the question in your title is "Yes".
However, as to the question of who was the last monarch to rule in South-West Africa, it is worth noting that, even as Head of State, the British monarch did not rule in South Africa (and hence South-West Africa under the League of Nations mandate).
Although Elizabeth II was titled Her Majesty Elizabeth the Second, Queen of South Africa and of Her other Realms and Territories, Head of the Commonwealth from 1953 to 1961, as a constitutional monarch, she reigned but did not rule.
Note that South-West Africa in the mandate period (1919-1966) was a very different case from, for example, the Aden Protectorate. South-West Africa was governed under a Class-C League of Nations mandate, not as a British Protectorate. Under that League of Nations mandate, the territory of South-West Africa became legally an "integral portion" of South Africa.
