Is my logic correct checking for a newer remote file?
Well, if it works, it works. There is not much to say about this small part of the code without any context. You should know this by yourself, because you should have some tests.



For the code:

The first catch is not suitable. If it happens, you will get a NullPointerException for the following if.



Avoid abbreviations. There are a lot of bad examples in this code, the most noticeable one is String e = s + " | Status Code: " + i;



Reduce indentation levels:

if (serverLastMod.after(fileLastFetched)) {
...
} else {
    Log.e("DS", "NO! Don't do anything... Wait until next check...");
}


could be:

if (!serverLastMod.after(fileLastFetched)) // or last &lt;= current
{
    log...
    return
}
... rest of code...


and so on.
The part ...rest of code... could probably have its own method update..., so the purpose is clear.