After Plan Matador had failed to be initiated (by Brooke-Popham) the British knew they sooner or later would see the Japanese at the gates of Singapore but very little if anything was done to prepare the defence. In the more than two months passing before the IJA reached Singapore a lot could have been done.
The Engineer officer of Singapore repeatedly asked Precival for permission to build field fortifications etc. to prepare the defence of Singapore but was told not to as this would only unease the civilian population (!).
Most of the guns of the Singapore Fortress could very well cover the north, but hardly any HE ammo was available, only AP for use against heavily armoured ships. A good commander would have seen to this in time.
The main problem of Singapore was that her fresh water supply depended on reservoirs north of the strait, but very little was done to prepare the defence of these, building up stocks or evacuating civilians. Considering the extreme strategic importance of Singapore no cost really would have been to high.
Against a prepared defence the first Japanese assaults probably would have failed as they were rather a spent ball once they reached Singapore. A lengthy siege would probably be very costly for the defenders incl. the civilians - Leningrad under the burning sun. But the Japanese would need time to get siege artillery and supplies forward and by that time the British will also have gathered reinforcement. The Brigades and landing crafts used in the OTL Madagascar operation (early April 42) would have given an interesting option for an outflanking landing on the Malayan westcoast.
Most of the British and CW troops committed in Malaya and Singapore indeed were relatively raw, but the regular ones actually committed themselves quite well. In a lenghty siege it would be possible to improve the raw troops (or at least use as labour) but also better utilise the regulars that in OTL arrived just before the surrender (18th Division). When fairly trained and equipped Indian troops aquitted themselves very well. The 4th Indian Division (prewar professional), which fought in the Mediterranean area from 1940, probably was one of the best Divisions on allied side in WWII. The 11th Indian Division in Singapore was made up of recruits and not fully equipped, and simply fell apart. Many of the PoWs joined the Japanese afterwards.
Regards
Steffen Redbeard