I wouldn’t worry about his umbilical cord at all, quite honestly that’s the least of his problems. Here is what Messybeast says about Premature kittens:
“Premature kittens happen when the mother goes into labour early for some reason, including physical trauma (some traumas may necessitate caesarian delivery). Many vets avoid late-pregnancy abortions because the kittens are viable. Females who birth early may fail to nurse the kittens because the hormones triggering maternal behaviour are not released; she may even kill the kittens as a result.
“Some pedigree breeds kitten two or three days early, without any problems. F1 hybrids of a domestic cat and a larger wild cat (e.g. serval) may be born on time as far as the mother cat is concerned, but at an earlier stage of kitten development because the wild species has a longer pregnancy. This means that the kittens are premature in terms of how well developed they are.
“My own experience of premmies is from late-aborted kittens (at the owner’s insistence) which were fostered by a shelter kitty who had given birth the same day. As with other orphan kittens, a foster mother is preferable as fostered kittens can suckle at will and get more maternal attention. Where not possible, a human foster mother is necessary.
“The more premature a kitten is, the less likely it is to survive. Survival chances decrease rapidly for kittens more than 5 days premature. Kittens born more than 2 weeks premature are unlikely to survive. They are small, fragile and easily become chilled. Their lungs will be poorly developed and lacking in surfactant (the moisture which lines the airways in the lungs). Their digestive systems will also be immature and unable to cope with full-strength kitten formula.
“It is very important that the kittens get queen’s colostrum in the first few hours of life. This provides maternal antibodies giving them passive immunity to a number of diseases. To prevent stressing their systems, their first feed should be glucose solution (they cannot store glucose for long and burn it up at a tremendous rate) and then diluted milk solution. It is best not to feed full strength milk until the kittens reach their due date. This reduces the strain on their immature digestive systems. They will probably not increase greatly in weight until they have passed their due date.
“Their sucking reflex may also be poor and they may require tube feeding. Kittens born more than 6 days early should be tube fed. Very premature kittens also require steroids and antibodies (administered by the vet) to help their immature lungs.
“Premature kittens often resemble mice in that their tails, legs and ears lack fur. Sometimes the body fur may be so fine as to be barely visible. In general, the first week or so of their lives is spent reaching the full-development stage at which they should have been born. They don’t put on weight at the same rate as a full term kitten because their bodies are expending energy on growing fur (up to 30% of their intake may be devoted to growing and maintaining fur) and “finishing off” immature internal organs.
“When the fur has “grown in” they often gain weight rapidly. Because premature kittens must finish off crucial development outside of the womb, weight gain often begins later than in full term kittens e.g. a week later, though they usually catch up after several weeks.
“A two week premature kitten will be at a similar stage of development to a normal few days old kitten, not at a similar stage to a two week old born at full term. I have found it helpful to think of them as “minus 2 weeks old”, “minus one week old” instead of “a week old”, “two weeks old” etc as this gives a more realistic idea of how fast they should be growing.”
It may be that your best chance with this little one is to get a vet to show you how to tube feed him.