What is the time it takes to measure the SINR in LTE?
So you're giving your answer yourself in the question:


  In general, how the standard LTE works to evaluate the SINR at the receiver?


The point is that measures like SNR and SINR are dependent on what you consider your signal of interest, what you consider interference and what you consider noise¹.

LTE actually specifies multiple different things that describe "received signal strength", whcih you can pu in relation to unwanted effects, and all of these could be labeled as SINR (but have different values for the same transmission!). The two "S" in ("SINR") measurements you probably care about are


RSRP, Reference Signal Received Power: Simply put, LTE uses distributed pilot symbols to figure out the channel state; just adding up the strengths of these will give you the magnitude of the impulse response over all OFDM subchannels.
RSRQ, Reference Signal Received Quality: The reference tone powers above don't say overly much about how well-suited the channel actually is – they are really just a sum. But in an OFDM system, where all but let's say 5% of subchannels are heavily attenuated, but these 5% are extremely well power-conserving, you just won't get the same performance in a system where all channels are attenuated, but things are more evenly distributed, even if the two systems have the same RSRP; hence, RSRQ takes other things into consideration, too.


You can (weightedly) subtract these from RSSI, which is also specificially defined for LTE, to get an estimate for the "IN" part of "SINR". 

Frankly, RSRQ and RSRP are probably much better suited to describe a complex multicarrier system than the simplistic SINR – so if I could, I'd just try to argue based on these, and drop the SINR terminology.


¹ This is actually the answer I most often give at my job, "SNR depends on what you consider to be signal and noise. No general formula can be given.".