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Is this route doable with the given time schedule (target time for being back from the Batu Caves in Kuala Lumpur: 15:00)?
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Yes - plenty of time.
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1.) How long is the travel time to Batu Caves?
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Not more than 1 hour in traffic from downtown KL (Golden Triangle/Bukit Bintang) - it's just a short distance away really, approx. 10 miles. Batu Caves is in the Sentul/Jalan Ipoh area north of KL.
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2.) Which options do I have to go there? (I'm interested in faster options.)
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Take the train from the airport to the KL Sentral station in Brickfields. Grab a cab from there.
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3.) How long do you recommend to stay at Batu Caves?
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Outside the main Hindu festivals (Thaipusam etc), Bat Caves is relatively deserted. I'd spend 45 minutes to an hour there looking at the limestones and the Hindu deities etc. Maybe take in some fresh coconut juice at the foot of the steps: street vendors there will hack open a fresh young green coconut open for you and watch monkeys scampering around.
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4.) Any recommendation for lunch/dinner appreciated (first class cuisine, excellent service, Asian cuisine preferred). The restaurant must open quite early because of my departure time. Lunch/Dinner at ~15:00 would be best.
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Sorry can't help much here - not much of a first class cuisine foodie. 15:00 is not a problem-food in KL is a round-the-clock thing. I'm of the opinion though the best KL has to offer isn't be in a posh restaurant.