Is there any state in the world, in which its Constitution doesn't call its President as the guarantor of the Constitution? Does "President" automatically imply that this person is also the guarantor of observance of the Constitution in the country?
While all elected politicians are expected to obey the law and respect the constitution, in Spain the guarantor of the constitution is the Tribunal Constitucional. This court is the one who examines laws made by the central ("federal") government and the autonomous regions ("states"), ruling if they're compatible with the constitution or not. Spain's head of state is its king, and he can (theoretically) refuse to sign any law, but since he's a mostly decorative figure, there's no actual way to veto a law, only denouncing it to the Constitutional Court and await its sentence.
There is at least one State, the United Kingdom, that has no formal Constitution - and no President, for that matter. So there's at least one country with no one formally in charge of guaranteeing its Constitution's observance.
Of the remaining countries, the Comparative Constitutions Project might have the relevant data to give a definite answer. Austria's President might be a good candidate, in that it has no veto power. But note that there are at least two enforcement mechanisms to guarantee that a Constitution is respected:
Countries can have someone with veto powers - usually a President or a Monarch. This role can involve defending the constitution, either because it's baked into the constitution itself (like in the US, whose Presidential oath's wording is part of the Constitution), or because of tradition (like in Germany). But it can also be entirely passive (like in the UK since 1704).
Countries can have an explicit Constitutional Court (like France) that can rip out part or all of a law on grounds that it doesn't respect the Constitution, or a High Court to the same effect (like the Supreme Court in the US).
Several countries which were British colonies have a president with a mostly ceremonial role, similar to the role of the Queen - lots of respect, hardly any influence. Such presidents can't be considered "guarantor of the Constitution".
The president of Ireland doesn't seem to have any powers over legislation (he may refer laws he considers unconstitutional to the supreme court, but must accept its ruling).
In Israel the president has a mainly ceremonial role with no actual power, certainly not "guarantor of the Constitution" (also because Israel doesn't have a constitution).
There are numerous states which do not name the president as the guarantor of their Constitution. For example, USA or Germany. But as a public servant, the president has to take an oath, and these oaths usually include something to the effect of supporting and upholding the Constitution (or, as a more general case, the law), so yes, it is more or less implied in the name of the post.
Does "President" automatically imply that this person is also the guarantor of observance of the Constitution in the country?
You've got it a bit backwards.
You see, the Constitution is a magic piece of paper that prevents politicians from doing bad things to the masses, by way of magic.
If something is explicitly forbidden by a Constitution, then politicians just can't do it.
For example, in the US, the Constitution prevents the mass-surveillance that's going- Oh wait.
Soviet constitutions declared certain political rights, such as freedom of speech, freedom of assembly, and freedom of religion.
In the Soviet Union, the Constitution "declared" freedom of speech, and so its magic powers prevented Stalin from doing bad stuff to people who wanted to voice their concerns!
You probably get the idea. Any Constitution is essentially a PR-statement by the government. It says things you like to hear, but it doesn't actually matter at all, so something like a "guarantor of observance" doesn't make sense.