The answer to the question, can I exercise the option right away? depends on the exercise style of the particular option contract you are talking about. If it's an American-style exercise, you can exercise at any moment until the expiration date. If it's an European-style exercise, you can only exercise at the expiration date.
According to the CME Group website on the FOPs on Gold futures, it's an American-style exercise (always make sure to double check this - especially in the Options on Futures world, there are quite a few that are European style): http://www.cmegroup.com/trading/metals/precious/gold_contractSpecs_options.html?optionProductId=192#optionProductId=192
So, if you wanted to, the answer is: yes, you can exercise those contracts before expiration.
But a very important question you should ask is: should you?
Some concepts first.
Option prices are composed of 2 parts: intrinsic value, and extrinsic value. Intrinsic value is defined as by how much the option is in the money. That is, for Calls, it's how much the strike is below the current underlying price; and for Puts, it's how much the strike is above the current underlying price. Extrinsic value is whatever amount you have to add to the intrinsic value, to get the actual price the option is trading at the market. Note that there's no negative intrinsic value. It's either a positive number, or 0. When the intrinsic value is 0, all the value of the option is extrinsic value.
The reason why options have extrinsic value is because they give the buyer a right, and the seller, an obligation. Ie, the seller is assuming risk. Traders are only willing to assume obligations/risks, and give others a right, if they get paid for that. The amount they get paid for that is the extrinsic value.
Analyzing your scenario.
In the scenario you described, underlying price is 1347, call strike is 1350. Whatever amount you have paid for that option is extrinsic value (because the strike of the call is above the underlying price, so intrinsic = 0, intrinsic + extrinsic = value of the option, by definition).
Now, in your scenario, gold prices went up to 1355. Now your call option is "in the money", that is, the strike of your call option is below the gold price. That necessarily means that your call option has intrinsic value. You can easily calculate how much: it has exactly $5 intrinsic value (1355 - 1350, undelrying price - strike). But that contract still has some "risk" associated to it for the seller: so it necessarily still have some extrinsic value as well.
So, the option that you bought for, let's say, $2.30, could now be worth something like $6.90 ($5 + a hypothetical $1.90 in extrinsic value).
How to make money out of that?
In your question, you mentioned exercising the option and then making a profit there. Well, if you do that, you exercise your options, get some gold futures immediately paying $1350 for them (your strike), and then you can sell them in the market for $1355. So, you make $5 there (multiplied by the contract multiplier). BUT your profit is not $5.
Here's why: remember that you had to buy that option? You paid some money for that. In this hypothetical example, you payed $2.30 to buy the option. So you actually made only $5 - $2.30 = $2.70 profit!
On the other hand, you could just have sold the option: you'd then make money by selling something that you bought for $2.30 that's now worth $6.90. This will give you a higher profit! In this case, if those numbers were real, you'd make $6.90 - $2.30 = $4.60 profit, waaaay more than $2.70 profit!
Here's the interesting part: did you notice exactly how much more profit you'd have by selling the option back to the market, instead of exercising it and selling the gold contracts? Exactly $1.90. Do you remember this number? That's the extrinsic value, and it's not a coincidence.
By exercising an option, you immediately give up all the extrinsic value it has. You are going to convert all the extrinsic value into $0. So that's why it's not optimal to exercise the contract. Also, many brokers usually charge you much more commissions and fees to exercise an option than to buy/sell options, so there's that as well!
Takeaways
Always remember: when you exercise an option contract, you immediately give up all the extrinsic value it has. So it's never optimal to do an early exercise of option contracts and individual, retail investors. (institutional investors doing HFT might be able to spot price discrepancies and make money doing arbitrage; but retail investors don't have the low commissions and the technology required to make money out of that!)
Might also be interesting to think about the other side of this: have you noticed how, in the example above, the option started with $2.30 of extrinsic value, and then it had less, $1.90 only? That's really how options work: as the market changes, extrinsic value changes, and as time goes by, extrinsic value usually decreases. Other factors might increase it (like, more fear in the market usually bring the option prices up), but the passage of time alone will decrease it.
So options that you buy will naturally decrease some value over time. The closer you are to expiration, the faster it's going to lose value, which kind of makes intuitive sense. For instance, compare an option with 90 days to expiration (DTE) to another with 10 DTE. One day later, the first option still has 89 DTE (almost the same as 90 DTE), but the other has 9 DTE - it relatively much closer to the expiration than the day before. So it will decay faster. Option buyers can protect their investment from time decay by buying longer dated options, which decay slower!
edit: just thought about adding one final thought here. Probabilities. The strategy that you describe in your question is basically going long an OTM call. This is an extremely bullish position, with low probability of making money. Basically, for you to make money, you need two things: you need to be right on direction, and you need to be right on time. In this example, you need the underlying to go up - by a considerable amount! And you need this to happen quickly, before the passage of time will remove too much of the extrinsic value of your call (and, obviously, before the call expires). Benefit of the strategy is, in the highly unlikely event of an extreme, unanticipated move of the underlying to the upside, you can make a lot of money. So, it's a low probability, limited risk, unlimited profit, extremely bullish strategy.
