1) First of all you could learn whole phrases or sentences with enough grammatical context to clarify the gender of the noun. Common ways would be to use pronouns -as @Schullz has already mentioned in his answer- or adjectives. MyLanguages.org gives an example of how in Slovak pronouns and adjectives change to agree with the grammatical gender of the noun:
While in English an adjective doesn’t change when the noun changes, in Slovak an adjective should agree in gender and number with the noun. For example:
a) Masculine to feminine and neutral example:
Toto je môj malý synček (this is my little son) becomes: Toto je moja malá dcerka (this is my little daughter). Toto je moje malé auto. (this is my little car).
As you can see from the example above, the adjective comes before the noun and also takes the feminine and neutral form.
[EDIT: Christoph Strobbe has mentioned in the comments that MyLanguages.org might not be entirely reliable. The general idea should hold though.]
So if for an example you are using a Spaced Repetion Software (Anki etc.) your cards could contain:
Q: my little son
A: môj malý synček
Or, if you prefer not to use any language except for your target language:
Q: synček (provide the standard phrase for learning the grammatical gender)
A: môj malý synček
In this case you would just have to remember that the standard phrase for learning the grammatical gender for all nouns is either môj malý, moja malá or moje malé.
2) As for congenitally blind people not being able to use Gabriel Wyner's mnemonic technique (or 'game') mentioned in the Answer you have linked to above - there is already a solution indirectly mentioned in the relevant quote from Wyner's book:
Masculine nouns burn. Feminine nouns are ice cold.
Feel the heat [of each word]; the more senses you can involve, the better.
(Emphasis added by me. I have also replaced the phrase 'of each image' with 'of each word', since a blind person would obviously have to make the kinesthetic associations with the words themselves, not with images.)
So congenitally blind people could use a non-visual version of Wyner's approach, based on the other senses.
In the light of this let's also take a look at another example for Wyner's approach, quoted in the answer you link to:
I want you to imagine all of the masculine nouns exploding. Your tree? Kaboom, splinters of wood everywhere. A branch gets embedded in the wall behind you. Dog chunks splatter all over the ceiling and floors. You wipe bits of fur and gore from your forehead. Make your images as vivid as you can stomach.
Even though congenitally blind people don't have the above-mentioned mental images they do have auditory, kinesthetic, olfactory and gustatory representations for the same things and experiences. They are therefore quite capable of mentally exploding a non-visual representation of a dog, having non-visual dog chunks splatter non-visually over the non-visual ceiling and floors, non-visually wiping non-visual bits of fur and gore from their forehead (even if most non-blind people couldn't really imagine what these mental representations would be like).
There are also many other non-visual ways of mentally associating nouns with grammatical gender: apart from the obvious example of mentally associating masculine nouns with a man's voice, feminine nouns with a woman's voice, and neuter nouns with a child's voice, one could even use voices from different actors, or a theatric, comic or tragic 'coloring' of the nouns to differentiate between the grammatical gender. All this would be a lot easier for a congenitally blind person, who is already accustomed to the exclusive use of non-visual mental representations. (This might also be a solution for some of those non-blind people who have difficulty applying visual mnemonics.)
To sum up: There is nothing hindering a congenitally blind person using a modified, non-visual version of Wyner's approach.