Learn English – Word for “the moment when you feel happy but deep down you are so scared to express that”

You can describe something as uplifting if it inspires happiness or hope. You can also hearten people.

Hearten - Make more cheerful or confident.

Such an object would be heartening.

Old is (generally) an adjective in English. So all its synonyms are likely to be adjectives as well.

Going off your original post, I would absolutely use "tired", "ancient", or "antiquated", depending on what you're trying to convey. Tired suggests it's long-since gone out of fashion but is still in use, while ancient suggests it's no longer in common usage, but was at one point. Antiquated further conveys that in addition to no longer being used, it's out of fashion.

Tired: hackneyed; stale: the same tired old jokes.

Ancient: Of, relating to, or belonging to times long past

Antiquated: Too old to be fashionable, suitable, or useful; outmoded.

Paired with "memes" (as you do in the question, with "old memes") this conveys exactly what I think you're getting at:

Oh no, grandpa saw a pack of my business cards, and added me to his hilarious e-mailing list, consisting of tired memes from 2007.

But if the one-word requirement is stronger than a requirement for common usage, "antiquities" might be sufficient. It doesn't convey the same condescending tone as tired or antiquated, but it does convey that something is dated:

Oh no, grandpa saw a pack of my business cards, and added me to his hilarious e-mailing list, consisting of antiquities from 2007.

Alternatively "crap" would convey that it's useless, but the reader has to infer that it's also dated by the "from 2007".

You'll notice that now the sentence doesn't clearly convey what is being emailed. I think that's going to be somewhat tricky to avoid with a single word because the condescension stems from pairing a derogatory or dismissive adjective with a normally neutral or positive noun like meme.

Neutral:

Grandpa sent me memes from 2007.

Condescending:

Grandpa sent me tired memes from 2007.

I don't believe "repost" is what you're looking for (though it is a reasonable single-word, if that's really your priority) because it doesn't necessarily convey age or staleness. In fact repost is often used for very new content that has simply been posted previously, and therefore this post isn't original. Often when one person sees a repost others are seeing for the first time. In that sense repost actually gives grandpa some credit, because perhaps the others on the chain haven't seen what he's sending.