I know about the Carbon dating but facilities are not available in our laboratory. Is any other laboratory methods for the study of fossil sample? How can I start my study on a fossil sample. Please suggest and guide about the procedure.
It depends. What kind of fossil you are talking about? Every absolute age estimation has its limits in material and method. A lot of geologists and palaeontologists would insist, that something that can be dated by C14 can not be called "fossil" because of the limited time range of this method.
Without more specific informations about your object of interest and issues you want to study nobody can give you specific answers to your questions.
there is no single generally accepted definition, at which time or which state of mineralisation an organism (or a group of them) can be called "fossil". The two most common ones used by palaeontologists and geologists are those:
With definition over the term "petrification", but with biomineralization (e.g. recent stromatolites) and early precipitation of Minerals (phospates within seconds after death, for example the fossil lagerstaetten of the Cretaceous Araripe Basin in Brazil, the Orsten in Scandinavia, but also only 100-year-old completley mineralized fishes from Greenland or freshly mineralized woods in mineshafts, or freshly dead organisms in some african lakes or islandic hot springs with highly mineralized waters) you can call a lot of living or allready died organisms "fossil", which is not really appropriate in my understanding, but "state of the art" among palaeontologists not dealing with quaternary geology or quat. palaeontology. With this definition an organism, still preserved but not mineralized (which is typical for most material which can be dated in the range of the C14-method) can not be called "fossil". Even not much older material like palynomorphs/acritarchs from the Cambrian, which are maybe alterated, but not mineralized. This definition, beside serious science, is unfortunately also fine for Kreationism.
With definition over timescales (and this is only an artificial category, too) a lot of people define (and also me, I have to confess) "fossil" as older than Holocene, which means also frosted reindeers from the LGM in Sibiria, with still eatable meat, are fossils. From my knowledge, this definition is also broadly accepted by a lot of palaeontologists, but some of them define it as "older than Quaternary" and with this definition all material datable by C14 is out (this was my suggestion, but not my personal point in my answer (on the upstanding very unprecise question)). In the opposite there are some followers of the anthropocene-concept going the other direction on the timeline, but this concept is less accepted under stratigraphers for good reasons.
Calling holocene organisms "fossil" (without taking even care about state of mineralization or time) is often more a terminology adaption by prehistoric archaeologist and bioscientists without ideas about the concepts of the term "fossil". I guess it is used in this way because it sounds better in public presentations and applications for funding, if they can copycat the "dinosaur-factor". The term "subfossil" is an extremly weak one between "recent" and "fossil" and not really specific. It should be avoided from my point of view.
Every one of this definitions has its weak points, but you have less discussion about "flexible" terms, if you use (more or less) clear cathegories. From this point of view, the definition over time (like in stratigraphy) is more likely than about taphonomy, which is very unprecise because of its wide range in timescales. Very weak from definition (but used also in peer-reviewed publications) is the approach of archaeologists and ecologists dealing with holocene material. I guess the reviewers of those articles use the term in the same undefined and "slang" way. Other approaches in definition are over genetics or morphometrics (e.g. the stupid term "living fossils"), if you like we can discuss this issues, too.
I guess, you will not find an actual definition, even in not "fossil" peer reviewed publications, everybody is fine with. From this point of view, maybe my answer is not getting the point you want it to, but you accidently forget to give the definition you have encountered in your question.
Dealing with Quaternay material, I prefer to use the terms of stratigraphy instead of "fossil", to avoid this minefield. But from the starting question it is not clear, from which timeslice in earth history the "fossils" are from, so its difficult to discuss one of the dating methods which are generally used to resolve such questions. (edit: And a question backto you: how many percent of time of the fossil record can be covered by the C14-Method? I think, following a rough estimation, it is something around 0.00625%. I guess that is wide under normal standard errors of statistics, and so on the end of the list of absolute dating methods regarding the starting question. )